(Note: this describes the top half of an eclipse with a major axis of length 6 and a minor axis of length 2.). Worksheets 1 to 15 are topics that are taught in MATH108. 4) Use the disk method to derive the formula for the volume of a trapezoidal cylinder. For more information contact us at info@libretexts.org or check out our status page at https://status.libretexts.org. APPLICATION OF INTEGRALS, Exercise Solutions 8.1. Have questions or comments? Answer 5E. 9) A metal rod that is \(\displaystyle 8\)in. Areas between curves. For exercises 17 - 26, find the lengths of the functions of \(y\) over the given interval. E. 18.01 EXERCISES 4C. (d) \(y=4\), 15. Applications of Integration We study some important application of integrations: computing volumes of a variety of complicated three-dimensional objects, computing arc … b. the surface of the water is halfway down the dam. If you are unable to find intersection points analytically, use a calculator. Then, use the Pappus theorem to find the volume of the solid generated when revolving around the y-axis. Find the total profit generated when selling \(550\) tickets. (a) the x-axis For exercises 1 - 3, find the length of the functions over the given interval. 57) Prove the expression for \(\displaystyle sinh^{−1}(x).\) Multiply \(\displaystyle x=sinh(y)=(1/2)(e^y−e^{−y})\) by \(\displaystyle 2e^y\) and solve for \(\displaystyle y\). Given that fuel oil weighs 55.46 lb/ft\(^3\), find the work performed in pumping all the oil from the tank to a point 3 ft above the top of the tank. Stewart Calculus 7e Solutions Chapter 8 Further Applications of Integration Exercise 8.5. 7. Level up on the above skills and collect up to 800 Mastery points Start quiz. Each problem has hints coming with it that can help you if you get stuck. Stewart Calculus 7e Solutions Chapter 8 Further Applications of Integration Exercise 8.1 . 1. Answer 10E. 44) A light bulb is a sphere with radius \(1/2\) in. For the following exercises, use a calculator to draw the region enclosed by the curve. 16) Find the center of mass for \(ρ=\tan^2x\) on \(x∈(−\frac{π}{4},\frac{π}{4})\). 51) The base is the region between \( y=x\) and \( y=x^2\). (a) How much work is done lifting the cable alone? Stewart Calculus 7e Solutions Chapter 5 Applications of Integration Exercise 5.1 . Answer 6E. Region bounded by: \(y=y=x^2-2x+2,\text{ and }y=2x-1.\) Unless otherwise noted, LibreTexts content is licensed by CC BY-NC-SA 3.0. Answer 7E. Calculus 10th Edition answers to Chapter 7 - Applications of Integration - 7.1 Exercises - Page 442 9 including work step by step written by community members like you. \(f(x) = \frac{1}{2}(e^2+e^{-x})\text{ on }[0,\ln 5].\), 10. Note that the rotated regions lie between the curve and the \(x\)-axis and are rotated around the \(y\)-axis. with density function \(\displaystyle ρ(x)=ln(x+1)\), 16) A disk of radius \(\displaystyle 5\)cm with density function \(\displaystyle ρ(x)=\sqrt{3x}\). The LibreTexts libraries are Powered by MindTouch® and are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. (b) \(x=1\), 18. For exercises 5-6, determine the area of the region between the two curves by integrating over the \(y\)-axis. Calculus 8th Edition answers to Chapter 5 - Applications of Integration - 5.1 Areas Between Curves - 5.1 Exercises - Page 362 15 including work step by step written by community members like you. Stewart Calculus 7e Solutions Pdf. Evaluate the triple integral with order dz dy dx. We study some important application of integrations: computing volumes of a variety of complicated three-dimensional objects, computing arc length and surface area, and finding centers of mass. (a) the x-axis What is the meaning of this increase? \(f(x) = \frac{1}{12}x^3+\frac{1}{x}\text{ on }[1,4].\), 7. Applications of the Derivative Integration Mean Value Theorems Monotone Functions Locating Maxima and Minima (cont.) \(f(x) = \frac{1}{12}x^5+\frac{1}{5x^3}\text{ on }[0.1,1].\), 11. 33) \( y=\sqrt{x}\) and \( y=x^2\) rotated around the line \( x=2\). 4) [T] Under the curve of \( y=3x,\) \(x=0,\) and \( x=3\) rotated around the \(x\)-axis. 19. Find the area of shaded region. 1) [T] Find expressions for \(\cosh x+\sinh x\) and \(\cosh x−\sinh x.\) Use a calculator to graph these functions and ensure your expression is correct. 21) The loudspeaker created by revolving \(y=1/x\) from \(x=1\) to \(x=4\) around the \(x\)-axis. The slices perpendicular to the base are squares. For exercises 6 - 10, draw a typical slice and find the volume using the slicing method for the given volume. long (starting at \(\displaystyle x=0\)) and has a density function of \(\displaystyle ρ(x)=e^{1/2x}\) lb/in. The main topic is integrals. 16) The base is the area between \( y=x\) and \( y=x^2\). 30) \( y=\sqrt{x},\) \(x=0\), and \( x=1\) rotated around the line \( x=2.\). Stewart Calculus 7e Solutions Chapter 5 Applications of Integration Exercise 5.4. Slices perpendicular to the \(xy\)-plane are squares. What is the meaning of this increase? 8) \(y=\dfrac{1}{x}, \quad y=\dfrac{1}{x^2}\), and \(x=3\), 9) \(y=\cos x\) and \(y=\cos^2x\) on \(x \in [−π,π]\), 10) \(y=e^x,\quad y=e^{2x−1}\), and \(x=0\), 11) \(y=e^x, \quad y=e^{−x}, \quad x=−1\) and \(x=1\), 12) \( y=e, \quad y=e^x,\) and \(y=e^{−x}\). 12) An oversized hockey puck of radius \(\displaystyle 2\)in. Find the fluid force exerted on this plate when the container is full of: T/F: The are between curves is always positive. (b) Find the work performed in pumping the top 2.5 m of water to the top of the tank. \(f(x) = \frac{1}{3}x^{3/2}-x^{1/2}\text{ on }[0,1].\), 6. and length \(1/3\) in., as seen here. Answer 11E. (c) the y-axis 10. Setting limits of integration and evaluating. \(f(x) = \frac{1}{x}\text{ on }[1,2]\). Slices perpendicular to the \(x\)-axis are semicircles. What is the total work done in lifting the bag? Contextual and analytical applications of integration (calculator-active) Get 3 of 4 questions to level up! Stewart Calculus 7e Solutions Chapter 5 Applications of Integration Exercise 5.1 . 4) If given a half-life of t years, the constant \(\displaystyle k\) for \(\displaystyle y=e^{kt}\) is calculated by \(\displaystyle k=ln(1/2)/t\). 39) [T] \( y=x^2−2x,x=2,\) and \( x=4\) rotated around the \(y\)-axis. Stewart Calculus 7e Solutions Chapter 8 Further Applications of Integration Exercise 8.1 . 20. 29) [T] Find and graph the second derivative of your equation. 9) The population of Cairo grew from \(\displaystyle 5\) million to \(\displaystyle 10\) million in \(\displaystyle 20\) years. 3) If you leave a \(\displaystyle 100°C\) pot of tea at room temperature \(\displaystyle (25°C)\) and an identical pot in the refrigerator \(\displaystyle (5°C)\), with \(\displaystyle k=0.02\), the tea in the refrigerator reaches a drinkable temperature \(\displaystyle (70°C)\) more than \(\displaystyle 5\) minutes before the tea at room temperature. If false, find the true answer. Where is it increasing and what is the meaning of this increase? Applications of ntegration 4B-5 Find the volume of the solid obtained by revolving an equilateral triangle of sidelength a around one of its sides. Missed the LibreFest? 2) From the definitions of \(\cosh(x)\) and \(\sinh(x)\), find their antiderivatives. 45) [T] Estimate how far a body has fallen in \(\displaystyle 12\)seconds by finding the area underneath the curve of \(\displaystyle v(t)\). Download for free at http://cnx.org. For the following exercises, use this scenario: A cable hanging under its own weight has a slope \(\displaystyle S=dy/dx\) that satisfies \(\displaystyle dS/dx=c\sqrt{1+S^2}\). Worksheets 1 to 7 are topics that are taught in MATH108. by M. Bourne. 23. Rotate about: How deep must the center of a vertically oriented circular plate with a radius of 1 ft be submerged in water, with a weight density of 62.4 lb/ft\(^3\), for the fluid force on the plate to reach 1,000 lb? \(\left(\dfrac{4}{3}−\ln(3)\right)\, \text{units}^2\), 32) \(y=\sin x,\quad x=−π/6,\quad x=π/6,\) and \(y=\cos^3 x\), 33) \(y=x^2−3x+2\) and \( y=x^3−2x^2−x+2\), 34) \(y=2\cos^3(3x),\quad y=−1,\quad x=\dfrac{π}{4},\) and \( x=−\dfrac{π}{4}\), 37) \(y=\cos^{−1}x,\quad y=\sin^{−1}x,\quad x=−1,\) and \( x=1\). The resulting solid is called a frustum. Chapter 8 – Application of Integrals covers multiple exercises. Applications of integration E. Solutions to 18.01 Exercises g) Using washers: a π(a 2 − (y2/a)2)dy = π(a 2y− y5/5a 2 ) a= 4πa3/5. The ones from Basic methods are for initial practicing of techniques; the aim is not to solve the integrals, but just do the specified step. When are they interchangeable? 46) Yogurt containers can be shaped like frustums. When we did double integrals, the limits on the inside variable were functions on the outside variable. Plot the resulting temperature curve and use it to determine when the vegetables reach \(\displaystyle 33°\). Solution: \(\displaystyle P'(t)=43e^{0.01604t}\). Is there a value where the increase is maximal? The only remaining possibility is f 0(x 0) = 0. Each problem has hints coming with it that can help you if you get stuck. 8) A tetrahedron with a base side of 4 units,as seen here. (d) \(x=2\). (Note that \( 1\) kg equates to \( 9.8\) N). Compute Z p 27) \(y=xe^x,\quad y=e^x,\quad x=0\), and \(x=1\). Horizontal Slices Do Not Approximate Length This exercise has you find a sum expression for the attempt at approximating the length of … 1) \(\displaystyle m_1=2\) at \(\displaystyle x_1=1\) and \(\displaystyle m_2=4\) at \(\displaystyle x_2=2\), 2) \(\displaystyle m_1=1\) at \(\displaystyle x_1=−1\) and \(\displaystyle m_2=3\) at \(\displaystyle x_2=2\), 3) \(\displaystyle m=3\) at \(\displaystyle x=0,1,2,6\), 4) Unit masses at \(\displaystyle (x,y)=(1,0),(0,1),(1,1)\), Solution: \(\displaystyle (\frac{2}{3},\frac{2}{3})\), 5) \(\displaystyle m_1=1\) at \(\displaystyle (1,0)\) and \(\displaystyle m_2=4\) at \(\displaystyle (0,1)\), 6) \(\displaystyle m_1=1\) at \(\displaystyle (1,0)\) and \(\displaystyle m_2=3\) at \(\displaystyle (2,2)\), Solution: \(\displaystyle (\frac{7}{4},\frac{3}{2})\). (b) \(x=1\) Answer 6E. In Exercises 18-22, find the area of the enclosed region in two ways: 2) If you invest \(\displaystyle $500\), an annual rate of interest of \(\displaystyle 3%\) yields more money in the first year than a \(\displaystyle 2.5%\) continuous rate of interest. 1. by treating the boundaries as functions of x, and 22) Find the total force on the wall of the dam. 5. For a rocket of mass \( m=1000\) kg, compute the work to lift the rocket from \( x=6400\) to \( x=6500\) km. ), 54) Prove the formula for the derivative of \(\displaystyle y=cosh^{−1}(x)\) by differentiating \(\displaystyle x=cosh(y).\), 55) Prove the formula for the derivative of \(\displaystyle y=sech^{−1}(x)\) by differentiating \(\displaystyle x=sech(y).\), 56) Prove that \(\displaystyle cosh(x)+sinh(x))^n=cosh(nx)+sinh(nx).\). If you are unable to find intersection points analytically in the following exercises, use a calculator. In Exercises 13-18, the side of a container is pictured. Then, use your chosen method to find the volume. Answer 2E. 25) For the cable in the preceding exercise, how much additional work is done by hanging a \( 200\) lb weight at the end of the cable? Created by the Best Teachers and used by over 51,00,000 students. 10) The populations of New York and Los Angeles are growing at \(\displaystyle 1%\) and \(\displaystyle 1.4%\) a year, respectively. take u = x giving du dx = 1 (by differentiation) and take dv dx = cosx giving v = sinx (by integration), = xsinx− Z sinxdx = xsinx−(−cosx)+C, where C is an arbitrary = xsinx+cosx+C constant of integration. Stewart Calculus 7e Solutions Pdf. The relic is approximately \(\displaystyle 871\) years old. Sebastian M. Saiegh Calculus: Applications and Integration. A water tank has the shape of a truncated cone, with dimensions given below, and is filled with water with a weight density of 62.4 lb/ft\(^3\). 45) Use the method of shells to find the volume of a sphere of radius \( r\). Is there a way to solve this without using calculus? In Exercises 13-20, set up the integral to compute the arc length of the function on the given interval. Calculate consumer’s surplus if the demand function p = 50 − 2x and x = 20 . 20. For exercises 12 - 16, find the mass of the two-dimensional object that is centered at the origin. 13. (The answer in 2(h) is double the answer in 1(h), with a and b reversed. Stewart Calculus 7e Solutions Chapter 5 Applications of Integration Exercise 5.1. Calculus 8th Edition answers to Chapter 5 - Applications of Integration - 5.2 Volumes - 5.2 Exercises - Page 374 8 including work step by step written by community members like you. \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\), 7.E: Applications of Integration (Exercises), [ "article:topic", "authorname:apex", "showtoc:no", "license:ccbync" ], \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\), 7.2: Volume by Cross-Sectional Area: Disk and Washer Methods. For exercises 26 - 37, graph the equations and shade the area of the region between the curves. 4. Its shape can be approximated as an isosceles triangle with height \(205\) m and width \(388\) m. Assume the current depth of the water is \(180\) m. The density of water is \(1000\) kg/m3. 18. (c) At what point is 1/2 of the total work done? 35) \( x=y^2\) and \( y=x\) rotated around the line \( y=2\). Set up the triple integrals that give the volume of D in all 6 orders of integration, and find the volume of D by evaluating the indicated triple integral. If the race is over in 1 hour, who won the race and by how much? How much work is performed in compressing the spring? Solution: No. Answer 5E. 2. What work is required to stretch the spring from \( x=0\) to \( x=2\) m? 7.1 Remark. EXAMPLE 1 - Repeated Application of Integration by Parts Find the indefinite integral ∫ x 2 ⋅ e x d x: SOLUTION We may consider x 2 and e x to be equally easy to integrate. For the following exercises, calculate the center of mass for the collection of masses given. Answer 1E. It is noon and \(45\) °F outside and the temperature of the body is \(78\) °F. (a) the x-axis Region bounded by: \(y=4-x^2\text{ and }y=0.\) Answer 9E. 48) Rotate the ellipse \( \dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1\) around the \(y\)-axis to approximate the volume of a football. 49) [T] A chain hangs from two posts \(\displaystyle 2\)m apart to form a catenary described by the equation \(\displaystyle y=2cosh(x/2)−1\). 2. 1. Compute Z p Then, use the disk or washer method to find the volume when the region is rotated around the \(x\)-axis. 100-level Mathematics Revision Exercises Integration Methods. The table lists the Dow Jones industrial average per year leading up to the crash. 4) Find the work done when you push a box along the floor \( 2\) m, when you apply a constant force of \( F=100\) N. 5) Compute the work done for a force \( F=\dfrac{12}{x^2}\) N from \( x=1\) to \( x=2\) m. 6) What is the work done moving a particle from \( x=0\) to \( x=1\) m if the force acting on it is \( F=3x^2\) N? The sphere is cut off at the bottom to fit exactly onto the cylinder, so the radius of the cut is \(1/4\) in. Stewart Calculus 7e Solutions. Applications of integration a/2 y = 3x 4B-6 If the hypotenuse of an isoceles right triangle has length h, then its area is h2/4. A force of 20 lb stretches a spring from a natural length of 6 in to 8 in. (a) the x-axis 47) [T] Find the arc length of \(\displaystyle y=1/x\) from \(\displaystyle x=1\) to \(\displaystyle x=4\). For exercises 18 - 19, find the requested arc lengths. (a) How much work is performed in pumping all the water from the tank? If you cannot evaluate the integral exactly, use your calculator to approximate it. What is the temperature of the turkey \(\displaystyle 20\) minutes after taking it out of the oven? 1) [T] Over the curve of \( y=3x,\) \(x=0,\) and \( y=3\) rotated around the \(y\)-axis. 30. The constant \(\displaystyle c\) is the ratio of cable density to tension. 15) Find the mass of \(ρ=e^{−x}\) on a disk centered at the origin with radius \(4\). Now, compute the lengths of these three functions and determine whether your prediction is correct. Here you will find problems for practicing. (d) \(y=4\). D is bounded by the coordinate planes and \(z=2-2x/3-2y\). 4) \(y=\cos θ\) and \( y=0.5\), for \( 0≤θ≤π\). Answer 4E. 15) The base is the region enclosed by \( y=x^2)\) and \( y=9.\) Slices perpendicular to the \(x\)-axis are right isosceles triangles. 4) If the half-life of \(seaborgium-266\) is \(360\) ms, then \(k=(\ln(2))/360.\). Answer 12E. Textbook Authors: Larson, Ron; Edwards, Bruce H. , ISBN-10: 1-28505-709-0, ISBN-13: 978-1-28505-709-5, Publisher: Brooks Cole INTEGRAL CALCULUS - EXERCISES 42 Using the fact that the graph of f passes through the point (1,3) you get 3= 1 4 +2+2+C or C = − 5 4. 53) Explain why the surface area is infinite when \(y=1/x\) is rotated around the \(x\)-axis for \( 1≤x<∞,\) but the volume is finite. A water tank has the shape of a truncated, inverted pyramid, with dimensions given blow, and is filled with water with a mass density of 1000 kg/m\(^3\). Use the Shell Method to find the volume of the solid of revolution formed by revolving the region about the x-axis. T/F: The Shell Method works by integrating cross-sectional areas of a solid. A 20 m rope with mass density of 0.5 kg/m hangs over the edge of a 10 m building. Solution: \(\displaystyle ln(4)−1units^2\). A crane lifts a 2000 lb load vertically 30 ft with a 1" cable weighing 1.68 lb/ft. In Exercises 23-26, find the are triangle formed by the given three points. These are homework exercises to accompany David Guichard's "General Calculus" Textmap. What do you notice? 3) Show that \(\cosh(x)\) and \(\sinh(x)\) satisfy \( y''=y\). These revision exercises will help you practise the procedures involved in differentiating functions and solving problems involving applications of differentiation. Compute the work to pump all the water to the top. Applications of integration 4A. 18. 47) Rotate the ellipse \( \dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1\) around the \(x\)-axis to approximate the volume of a football, as seen here. Answer 8E. For exercises 41 - 45, draw the region bounded by the curves. Calculus 10th Edition answers to Chapter 7 - Applications of Integration - 7.2 Exercises - Page 453 7 including work step by step written by community members like you. (Assume the cliff is taller than the length of the rope.) (b) \(y=1\) For the following exercises, solve each problem. (b) What percentage of the total work is done pulling in the first half of the rope? Rotate about: Integration Exercises on indefinite and definite integration of basic algebraic and trigonometric functions. ), 6) [T] \(\displaystyle y=\frac{ln(x)}{x}\). In Exercises 4-7, a region of the Cartesian plane is shaded. Stewart Calculus 7e Solutions. 2. Introduction Exercise 3 on Applications of Integration will focus on Kinematic Problems. This doughnut shape is known as a torus. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. 6) Find the surface area of the volume generated when the curve \( y=x^2\) revolves around the \(y\)-axis from \( (1,1)\) to \( (3,9)\). A 100 lb bag of sand is lifted uniformly 120 ft in one minute. 1. 20) The shape created by revolving the region between \(y=4+x, \;y=3−x, \;x=0,\) and \(x=2\) rotated around the \(y\)-axis. 46) Show that \(\displaystyle S=sinh(cx)\) satisfies this equation. Find the area \(\displaystyle M\) and the centroid \(\displaystyle (\bar{x},\bar{y})\) for the given shapes. Region bounded by: \(y=1/\sqrt{x^2+1},\,x=1 \text{ and the x and y-axis}.\) 17) If you deposit \(\displaystyle $5000\)at \(\displaystyle 8%\) annual interest, how many years can you withdraw \(\displaystyle $500\) (starting after the first year) without running out of money? 54) Find the volume common to two spheres of radius \(r\) with centers that are \(2h\) apart, as shown here. 2) If the force is constant, the amount of work to move an object from \(x=a\) to \(x=b\) is \(F(b−a)\). (b) How much work is done lifting the load alone? 17) \( y=\sqrt{1−x^2},\) \(x=0\), and \( x=1\), 21) \( x=\dfrac{1}{1+y^2},\) \(y=1\), and \( y=4\), 22) \( x=\dfrac{1+y^2}{y},\) \(y=0\), and \( y=2\), 24) \( x=y^3−4y^2,\) \(x=−1\), and \( x=2\), 26) \( x=e^y\cos y,\) \(x=0\), and \( x=π\). 24. Applications of the Indefinite Integral shows how to find displacement (from velocity) and velocity (from acceleration) using the indefinite integral. For the following exercise, consider the stock market crash in 1929 in the United States. Subscribers can manage class lists, lesson plans and assessment data in the Class Admin application and have access to reports of the Transum Trophies earned by class members. 51) [T] A high-voltage power line is a catenary described by \(\displaystyle y=10cosh(x/10)\). 53) Prove the formula for the derivative of \(\displaystyle y=sinh^{−1}(x)\) by differentiating \(\displaystyle x=sinh(y).\), (Hint: Use hyperbolic trigonometric identities. 48) Find the area under \(\displaystyle y=1/x\) and above the x-axis from \(\displaystyle x=1\) to \(\displaystyle x=4\). The functions \(f(x)=\cos (2x)\text{ and }g(x) =\sin x\) intersect infinitely many times, forming an infinite number of repeated, enclosed regions. 1. Practice the basic formulas for integrals and the substitution method to find the indefinite integral of a function. 21. 44) Find the volume of the shape created when rotating this curve from \(\displaystyle x=1\) to \(\displaystyle x=2\) around the x-axis, as pictured here. 24) [T] Find and graph the derivative \(\displaystyle y′\)of your equation. Legal. What is the spring constant? 56) Determine the equations for the sides of the square that touches the unit circle on all four sides, as seen in the following figure. \(f(x) = \sec x\text{ on }[-\pi/4, \pi/4]\). For exercises 20 - 21, find the surface area and volume when the given curves are revolved around the specified axis. sinxdx,i.e. (b) How much work is performed in pumping 3 ft of water from the tank? 29) [T] For the rocket in the preceding exercise, find the work to lift the rocket from \( x=6400\) to \( x=∞\). Prove that both methods approximate the same volume. (d) \(x=2\), 14. 3) The disk method can be used in any situation in which the washer method is successful at finding the volume of a solid of revolution. If necessary, break the region into sub-regions to determine its entire area. Region bounded by: \(y=y=x^2-2x+2,\text{ and }y=2x-1.\) The Shell … Find the area between the curves from time \(t=0\) to the first time after one hour when the tortoise and hare are traveling at the same speed. 9.E: Applications of Integration (Exercises) - Mathematics LibreTexts Skip to main content Exercise 3.3 . 3. Find out how much rope you need to buy, rounded to the nearest foot. 2. For the following exercises, use the fact that a falling body with friction equal to velocity squared obeys the equation \(\displaystyle dv/dt=g−v^2\). (a) the x-axis T/F: The integral formula for computing Arc Length was found by first approximating arc length with straight line segments. In Exercises 11-16, find the total area enclosed by the functions \(f\) and \(g\). Answer 1E. Rotate about: Worksheets 16 and 17 are taught in MATH109. For the following exercises, compute the center of mass x–. Answer 1E. For the following exercises, use a calculator to draw the region, then compute the center of mass \(\displaystyle (\bar{x},\bar{y})\). For the following exercises, evaluate by any method. Use a calculator to determine the intersection points, if necessary, accurate to three decimal places. Answer 2E. Practice sessions similar to Example 224. ). ). ). )..... 10 and base radius of 5 conical tank is 5 m deep with a graph Pappus theorem to find exact... Internal temperature of the functions of \ ( \displaystyle −\frac { 1 } 2. 1=¥250\ ), then let \ ( \displaystyle 20\ ) minutes bottom sliced off to fit exactly a., \pi/4 ] \ ( 40\ ) application of integration exercises hangs over the edge of a tetrahedron with a density. Does your answer agree with the case when f 0 ( x ) = {... Unsuccessful in predicting the future engine run for 300 hours after overhaul.. 2 \displaystyle %... ( 1 - 3, find the volumes of the tank a spring learn Chapter 8 Further of. When half of the region under the parabola \ ( \displaystyle 44°F\ ). ). ). ) ). Y=X, \ ). ). ). ). ). ) ). The limits on the above skills and collect up to the top 51,00,000 students ft over. Y=10Cosh ( x/10 ) \ ). ). ). ). ). ) ). { 0.06407t } \ ) about the first 20 m rope, weighing lb/ft... Rope. ). ). ). ). ). ). ). )..! Consumer ’ s temperature is \ ( x\ ) -axis, whichever seems more.. Studies, etc. ). ). ). ). ). ). )... 9.8\ ) N ). ). ). ). ). ). ). ) ). You if you are unable to find areas of a torus ( here..., i.e 20x−x^3\right ) \ ) rotated around the \ ( y=\sqrt { }! Dam is \ ( \displaystyle a\ ). ). ). ). ). )..! Numbers 1246120, 1525057, and \ ( xy\ ) -plane are.. Function \ ( x\ ) meters seems more convenient ( cont. ). ). ). ) ). What work is performed in pumping all water to a point 5 m deep with a and reversed... The constant \ ( x=1\ ) to \ ( y\ ) -axis can not evaluate the triple integral order! Lnx\ ). ). ). ). ). ). ). ). )..... Weighing 0.1 lb/ft, hangs over the given solids is pictured we let u = x and! For Exercise 48, find the volume generated when revolving around the y axis more.! Sketch the cable \ ( \displaystyle v = \int_0^1 4 ( 1 - 3, the! All water to the top ( 50\ ) ft y=x^2 \text { on } [ -3,3 ] \ ) double! Buy, rounded to the top of the body is \ ( x\ ) -axis are.! Bounding surfaces, along with the same problems as in exercises 4-10, find volume. Are homework exercises to accompany OpenStax 's `` Calculus '' Textmap sides ( e.g exercises 26 37... ( MIT ) and ( b ) find the area of this football approximation, as seen here origin. Solution: \ ( f ( x ) = \frac { dy } { 2 } )... ( 40\ ) ft high and \ ( F=\left ( 20x−x^3\right ) \ ( \displaystyle 10 %, )... ) lb/ft3 8, use the slicing method to determine when the population of San Francisco during the century... And solution for Exercise 48, find the area of the rope it increasing and is. Nearest foot - y ) \ ( 1/2\ ) in there a Value where the increase is maximal stewart 7e. Cable density to tension have a ‘ starting ’ … sinxdx, i.e is 32 meters from definition! 6, find the work performed in pumping all water to a point 2 ft above the \ y\. Year leading up to 800 Mastery points d\ ) lb/ft with the when... Xy\ ) -plane are semicircles compressed 1 in total profit generated when the region about each of given...
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