For exercises 3 - 4, split the region between the two curves into two smaller regions, then determine the area by integrating over the \(x\)-axis. Textbook Authors: Stewart, James , ISBN-10: 1285740629, ISBN-13: 978-1-28574-062-1, Publisher: Cengage 26. True or False? It is noon and \(45\) °F outside and the temperature of the body is \(78\) °F. Answer 11E. Answer 7E. 5) [T] Under the curve of \( y=2x^3,x=0,\) and \( x=2\) rotated around the \(y\)-axis. The solid formed by revolving \(y=2x \text{ on }[0,1]\) about the x-axis. If interest is a continuous \(\displaystyle 10%,\) how much do you need to invest initially? In Exercises 3-12, find the arc length of the function on the given interval. 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. Enjoy the videos and music you love, upload original content, and share it all with friends, family, and the world on YouTube. This page contains a list of commonly used integration formulas with examples,solutions and exercises. Rotate about: If each of the workers, on average, lifted ten 100-lb rocks \( 2\)ft/hr, how long did it take to build the pyramid? 43) Find the area between \(\displaystyle lnx\) and the x-axis from \(\displaystyle x=1\) to \(\displaystyle x=2\). exercises so that they become second nature. Chapter 8 – Application of Integrals covers multiple exercises. A velociraptor 64 meters away spots you. Answer 11E. Rotate the line \( y=\left(\frac{1}{m}\right)x\) around the \(y\)-axis to find the volume between \( y=a\) and \( y=b\). 41) \( y=\sqrt{x},\quad x=4\), and \( y=0\), 42) \( y=x+2,\quad y=2x−1\), and \( x=0\), 44) \( x=e^{2y},\quad x=y^2,\quad y=0\), and \( y=\ln(2)\), \(V = \dfrac{π}{20}(75−4\ln^5(2))\) units3, 45) \( x=\sqrt{9−y^2},\quad x=e^{−y},\quad y=0\), and \( y=3\). 2) Use the slicing method to derive the formula for the volume of a cone. 27) [T] For the pyramid in the preceding exercise, assume there were \( 1000\) workers each working \( 10\) hours a day, \( 5\) days a week, \( 50\) weeks a year. Use the Shell Method to find the volume of the solid of revolution formed by revolving the region about the y-axis. Stewart Calculus 7e Solutions Chapter 5 Applications of Integration Exercise 5.1 . 27. 14) Below \(x^2+y^2=1\) and above \(y=1−x\). Sand leaks from the bag at a rate of 1/4 lb/s. (Hint: Integration By Parts will be necessary, twice. Then, use the Pappus theorem to find the volume of the solid generated when revolving around the y-axis. For the following exercises, find the derivatives of the given functions and graph along with the function to ensure your answer is correct. What is the temperature of the turkey \(\displaystyle 20\) minutes after taking it out of the oven? Slices perpendicular to the \(y\)-axis are squares. In Exercises 13-18, the side of a container is pictured. For exercises 35 - 42, find the surface area of the volume generated when the following curves revolve around the \(y\)-axis. T/F: The integral formula for computing Arc Length was found by first approximating arc length with straight line segments. 31) \( y=\dfrac{1}{4−x},\) \(x=1,\) and \( x=2\) rotated around the line \( x=4\). \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\), 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. 48) \( y=\ln(\sin x)\) from \( x=π/4\) to \( x=(3π)/4\). The population is always increasing. 4) Use the disk method to derive the formula for the volume of a trapezoidal cylinder. Answer 2E. The cost of over haul of an engine is ₹10,000 The operating cost per hour is at the rate of 2x − 240 where the engine has run x km. Introduction 2 2. Stewart Calculus 7e Solutions Chapter 8 Further Applications of Integration Exercise 8.5 . 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. You know these dinosaurs lived during the Cretaceous Era (\(\displaystyle 146\) million years to \(\displaystyle 65\) million years ago), and you find by radiocarbon dating that there is \(\displaystyle 0.000001%\) the amount of radiocarbon. (c) the x-axis Solution: \(\displaystyle 239,179\) years. If you are unable to find intersection points analytically, use a calculator. EduRev, the Education Revolution! Stewart Calculus 7e Solutions Chapter 8 Further Applications of Integration Exercise 8.1 . For the following exercises, solve each problem. In Exercises 9-12, a region of the Cartesian plane is shaded. The water is to be pumped to a point 2 ft above the top of the tank. 53) Prove the formula for the derivative of \(\displaystyle y=sinh^{−1}(x)\) by differentiating \(\displaystyle x=sinh(y).\), (Hint: Use hyperbolic trigonometric identities. sinxdx,i.e. (b) \(x=2\) Answer 6E. For exercises 41 - 45, draw the region bounded by the curves. E. 18.01 EXERCISES 4C. For exercises 56 - 57, solve using calculus, then check your answer with geometry. Starting from \(\displaystyle $1=¥250\), when will \(\displaystyle $1=¥1\)? Start Unit test . What work is required to stretch the spring from \( x=0\) to \( x=2\) m? Find the area of the region bounded by x^2 = 4y, y = 2, y = 4 and the y-axis in the first quadrant. 52) A telephone line is a catenary described by \(\displaystyle y=acosh(x/a).\) Find the ratio of the area under the catenary to its arc length. 25) [T] Find and graph the second derivative of your equation. For exercises 1 - 3, find the length of the functions over the given interval. long (starting at \(\displaystyle x=5\)) and has a density function of \(\displaystyle ρ(x)=ln(x)+(1/2)x^2\) oz/in. Region bounded by: \(y=2x,\,y=x\text{ and }x=2.\) 21) A shock absorber is compressed 1 in. 1. (c) the x-axis 25) Find the volume of the catenoid \(y=\cosh(x)\) from \(x=−1\) to \(x=1\) that is created by rotating this curve around the \(x\)-axis, as shown here. Sebastian M. Saiegh Calculus: Applications and Integration. 17. (b) \(y=1\) Textbook Authors: Stewart, James , ISBN-10: 1285740629, ISBN-13: 978-1-28574-062-1, Publisher: Cengage 36) \( y=\frac{1}{2}x^2+\frac{1}{2}\) from \( x=0\) to \( x=1\), 38) [T] \( y=\dfrac{1}{x}\) from \( x=\dfrac{1}{2}\) to \( x=1\), 39) \( y=\sqrt[3]{x}\) from \( x=1\) to \( x=27\), 40) [T] \( y=3x^4\) from \( x=0\) to \( x=1\), 41) [T] \( y=\dfrac{1}{\sqrt{x}}\) from \( x=1\) to \( x=3\), 42) [T] \( y=\cos x\) from \( x=0\) to \( x=\frac{π}{2}\). For the following exercises, find the antiderivatives for the given functions. For the following exercises, compute the center of mass \(\displaystyle (\bar{x},\bar{y})\). Source: http:/www.factmonster.com/ipka/A0762181.html. For the following exercises, verify the derivatives and antiderivatives. Answer 10E. A = 2 −1 (x 2 + 1 − x)dx = (x 3 /3 + x − x 2 /2) 2 −1 = 9/2 2. Math exercises on integral of a function. 2. Exercise 3.2 . If you cannot evaluate the integral exactly, use your calculator to approximate it. 11) [T] \(\displaystyle \frac{1}{cosh(x)}\), Solution: \(\displaystyle −tanh(x)sech(x)\), 13) [T] \(\displaystyle cosh^2(x)+sinh^2(x)\), Solution: \(\displaystyle 4cosh(x)sinh(x)\), 14) [T] \(\displaystyle cosh^2(x)−sinh^2(x)\), 15) [T] \(\displaystyle tanh(\sqrt{x^2+1})\), Solution: \(\displaystyle \frac{xsech^2(\sqrt{x^2+1})}{\sqrt{x^2+1}}\), 16) [T] \(\displaystyle \frac{1+tanh(x)}{1−tanh(x)}\), Solution: \(\displaystyle 6sinh^5(x)cosh(x)\), 18) [T] \(\displaystyle ln(sech(x)+tanh(x))\). 6) \(y=x^2−x\), from \(x=1\) to \(x=4\), rotated around the \(y\)-axis using the washer method, 7) \(x=y^2\) and \(x=3y\) rotated around the \(y\)-axis using the washer method, 8) \(x=2y^2−y^3,\; x=0\),and \(y=0\) rotated around the x-axis using cylindrical shells, b.the volume of the solid when rotated around the \(x\)-axis, and. (Hint: all cross-sections are circles.). 1. water, with a weight density of 62.4 lb/ft\(^3\) 12) The effect of advertising decays exponentially. with density function \(\displaystyle ρ(x)=ln(x+1)\), 16) A disk of radius \(\displaystyle 5\)cm with density function \(\displaystyle ρ(x)=\sqrt{3x}\). Then, use the disk or washer method to find the volume when the region is rotated around the \(x\)-axis. 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. After reading this text, and/or viewing the video tutorial on this topic, you should be able to: • carry out integration by making a substitution • identify appropriate substitutions to make in order to evaluate an integral Contents 1. Applications of Integration. 23) You are a crime scene investigator attempting to determine the time of death of a victim. 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 5 Applications of Integration Exercise 5.4. Up next for you: Unit test. Do no evaluate the integral. by a force of \( 75\) lb. Solution: \(\displaystyle −\frac{1}{x(lnx)^2}\). Stewart Calculus 7e Solutions Chapter 8 Further Applications of Integration Exercise 8.1 . We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. In Exercises 23-26, find the are triangle formed by the given three points. Note that you will have two integrals to solve. Answer 11E. What is the total work done in lifting the box and sand? \(v(t) =-32t+20\)ft/s on [0,5]. How much work is done in lifting the box 1.5 ft (i.e, the spring will be stretched 1 ft beyond its natural length)? 4C-2 Find the vo Applications of Integration We study some important application of integrations: computing volumes of a variety of complicated three-dimensional objects, computing arc … Calculus 10th Edition answers to Chapter 7 - Applications of Integration - 7.1 Exercises - Page 442 17 including work step by step written by community members like you. For the following exercises, find the indefinite integral. Each problem has hints coming with it that can help you if you get stuck. For the following exercises, use this scenario: A cable hanging under its own weight has a slope \(\displaystyle S=dy/dx\) that satisfies \(\displaystyle dS/dx=c\sqrt{1+S^2}\). Does this confirm your answer for the previous question? 7) \( y=x^{3/2}\) from \( (0,0)\) to \( (1,1)\), 8) \( y=x^{2/3}\) from \( (1,1)\) to \( (8,4)\), 9) \( y=\frac{1}{3}(x^2+2)^{3/2}\) from \( x=0\) to \( x=1\), 10) \( y=\frac{1}{3}(x^2−2)^{3/2}\) from \( x=2\) to \( x=4\), 11) [T] \( y=e^x\) on \( x=0\) to \( x=1\), 12) \( y=\dfrac{x^3}{3}+\dfrac{1}{4x}\) from \( x=1\) to \( x=3\), 13) \( y=\dfrac{x^4}{4}+\dfrac{1}{8x^2}\) from \( x=1\) to \( x=2\), 14) \( y=\dfrac{2x^{3/2}}{3}−\dfrac{x^{1/2}}{2}\) from \( x=1\) to \( x=4\), 15) \( y=\frac{1}{27}(9x^2+6)^{3/2}\) from \( x=0\) to \( x=2\), 16) [T] \( y=\sin x\) on \( x=0\) to \( x=π\). 58) Prove the expression for \(\displaystyle cosh^{−1}(x).\) Multiply \(\displaystyle x=cosh(y)=(1/2)(e^y−e^{−y})\) by \(\displaystyle 2e^y\) and solve for \(\displaystyle y\). 23. 30. 12) An oversized hockey puck of radius \(\displaystyle 2\)in. 3. (a) the x-axis A force of 20 lb stretches a spring from a natural length of 6 in to 8 in. \((-1,1),\, (1,3)\text{ and }(2,-1)\). 22) A force of \( F=\left(20x−x^3\right)\) N stretches a nonlinear spring by \( x\) meters. \(f(x) = \sqrt{1-x^2}\text{ on }[-1,1].\)(Note: this describes the top half of a circle with radius 1. 39) Find the generalized center of mass between \(\displaystyle y=bsin(ax), x=0,\) and \(\displaystyle x=\frac{π}{a}.\) Then, use the Pappus theorem to find the volume of the solid generated when revolving around the y-axis. 5) The volume that has a base of the ellipse \(\dfrac{x^2}{4}+\dfrac{y^2}{9}=1\) and cross-sections of an equilateral triangle perpendicular to the \(y\)-axis. Chapter 7: Applications of Integration Course 1S3, 2006–07 May 11, 2007 These are just summaries of the lecture notes, and few details are included. 47) [T] You are building a bridge that will span \( 10\) ft. You intend to add decorative rope in the shape of \( y=5|\sin((xπ)/5)|\), where \( x\) is the distance in feet from one end of the bridge. (d) \(y=2\), 16. Source: http:/www.sfgenealogy.com/sf/history/hgpop.htm. 13) The base is the region under the parabola \( y=1−x^2\) in the first quadrant. 7) A wire that is \(\displaystyle 2\)ft long (starting at \(\displaystyle x=0\)) and has a density function of \(\displaystyle ρ(x)=x^2+2x\) lb/ft, 8) A car antenna that is \(\displaystyle 3\) ft long (starting at \(\displaystyle x=0)\) and has a density function of \(\displaystyle ρ(x)=3x+2\) lb/ft. \(f(x) = \frac{1}{3}x^{3/2}-x^{1/2}\text{ on }[0,1].\), 6. 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? 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