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In your own words, explain how the Disk and Washer Methods are related. For the following exercises, compute the center of mass \(\displaystyle (\bar{x},\bar{y})\). We use cross-sectional area to compute volume. 7.1 Remark. Chapter 14 - APPLICATIONS of INTEGRATION 333 The next exercise has you show what can go wrong when an “approximation” is not accurate. 24. (Hint: all cross-sections are circles.). 27) \(\displaystyle y=\sqrt{x^2+1}\sqrt{x2^−1}\), Solution: \(\displaystyle \frac{2x^3}{\sqrt{x^2+1}\sqrt{x^2−1}}\), Solution: \(\displaystyle x^{−2−(1/x)}(lnx−1)\), 33) \(\displaystyle y=\sqrt{x}\sqrt[3]{x}\sqrt[6]{x}\), Solution: \(\displaystyle −\frac{1}{x^2}\). (a) the x-axis 30) [T] A rectangular dam is \(40\) ft high and \(60\) ft wide. Answer 4E. 21) The loudspeaker created by revolving \(y=1/x\) from \(x=1\) to \(x=4\) around the \(x\)-axis. 17) A \( 12\)-in. A force of 20 lb stretches a spring from a natural length of 6 in to 8 in. Solution: \(\displaystyle P'(t)=43e^{0.01604t}\). Source: http:/stockcharts.com/freecharts/hi...a19201940.html. 6. 36) A cone-shaped tank has a cross-sectional area that increases with its depth: \( A=\dfrac{πr^2h^2}{H^3}\). Textbook Authors: Larson, Ron; Edwards, Bruce H. , ISBN-10: 1-28505-709-0, ISBN-13: 978-1-28505-709-5, Publisher: Brooks Cole What do you notice? The solid formed by revolving \(y=\sqrt{x} \text{ on }[0,1]\) about the x-axis. 22. For exercises 1 - 3, find the length of the functions over the given interval. How much work would it take to stretch the spring from \( 15\) cm to \( 20\) cm? You check on your vegetables \(\displaystyle 2\) hours after putting them in the refrigerator to find that they are now \(\displaystyle 12°F\). 26) Find surface area of the catenoid \(y=\cosh(x)\) from \(x=−1\) to \(x=1\) that is created by rotating this curve around the \(x\)-axis. 2. by treating the boundaries as functions of y. Unless otherwise noted, LibreTexts content is licensed by CC BY-NC-SA 3.0. In primary school, we learned how to find areas of shapes with straight sides (e.g. Then, use the washer method to find the volume when the region is revolved around the \(y\)-axis. (a) \(x=2\) 18) \( y=2x^2,\quad x=0,\quad x=4,\) and \( y=0\), \(\displaystyle V = \int_0^4 4\pi x^4\, dx \quad=\quad \frac{4096π}{5}\) units3, 19) \( y=e^x+1,\quad x=0,\quad x=1,\) and \( y=0\), \(\displaystyle V = \int_0^1 \pi\left( 1^2 - \left( x^4\right)^2\right)\, dx = \int_0^1 \pi\left( 1 - x^8\right)\, dx \quad = \quad \frac{8π}{9}\) units3, 21) \( y=\sqrt{x},\quad x=0,\quad x=4,\) and \( y=0\), 22) \( y=\sin x,\quad y=\cos x,\) and \( x=0\), \(\displaystyle V = \int_0^{\pi/4} \pi \left( \cos^2 x - \sin^2 x\right) \, dx = \int_0^{\pi/4} \pi \cos 2x \, dx \quad=\quad \frac{π}{2}\) units3, 23) \( y=\dfrac{1}{x},\quad x=2\), and \( y=3\), 24) \( x^2−y^2=9\) and \( x+y=9,\quad y=0\) and \( x=0\). T/F: The integral formula for computing Arc Length was found by first approximating arc length with straight line segments. 31) A sphere created by rotating a semicircle with radius \(\displaystyle a\) around the \(\displaystyle y\)-axis. Therefore, we let u = x 2 and write the following. 27) \( y=\sqrt{x}\) from \( x=2\) to \( x=6\), 30) [T] \( y=\frac{1}{x^2}\) from \( x=1\) to \( x=3\), 31) \( y=\sqrt{4−x^2}\) from \( x=0\) to \( x=2\), 32) \( y=\sqrt{4−x^2}\) from \( x=−1\) to \( x=1\), 34) [T] \( y=\tan x\) from \( x=−\frac{π}{4}\) to \( x=\frac{π}{4}\). 2.Find the area of the region bounded by y^2 = 9x, x=2, x =4 and the x axis in the first quadrant. In Exercises 9-12, a region of the Cartesian plane is shaded. These are homework exercises to accompany David Guichard's "General Calculus" Textmap. Answer 10E. In Exercises 5-8, a region of the Cartesian plane is shaded. For exercises 27 - 36, find the volume generated when the region between the curves is rotated around the given axis. 4. Answer 9E. Exercise 3.3 . Each problem has hints coming with it that can help you if you get stuck. 1) [T] Over the curve of \( y=3x,\) \(x=0,\) and \( y=3\) rotated around the \(y\)-axis. 22) Find the total force on the wall of the dam. Does your answer agree with the volume of a cone? Solutions to exercises 15 Exercise 2. If each of the workers, on average, lifted ten 100-lb rocks \( 2\)ft/hr, how long did it take to build the pyramid? For exercises 26 - 37, graph the equations and shade the area of the region between the curves. 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. 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. 26. Is this bone from the Cretaceous? 33) \( y=x+2,\quad y=x+6,\quad x=0\), and \( x=5\), 37) [T] \( y=\cos x,\quad y=e^{−x},\quad x=0\), and \( x=1.2927\), 39) \( y=\sin x,\quad y=5\sin x,\quad x=0\) and \( x=π\), 40) \( y=\sqrt{1+x^2}\) and \( y=\sqrt{4−x^2}\). Answer 5E. For exercises 53 - 55, find the area between the curves by integrating with respect to \(x\) and then with respect to \(y\). A skew right circular cone with height of 10 and base radius of 5. For exercises 7 - 13, graph the equations and shade the area of the region between the curves. What is the spring constant? T/F: A solid of revolution is formed by revolving a shape around an axis. For exercises 37 - 44, use technology to graph the region. For more information contact us at info@libretexts.org or check out our status page at https://status.libretexts.org. Do you obtain the same answer? Introduction 2 2. with density function \(\displaystyle ρ(x)=ln(x+1)\), 16) A disk of radius \(\displaystyle 5\)cm with density function \(\displaystyle ρ(x)=\sqrt{3x}\). Rotate about: Unless otherwise noted, LibreTexts content is licensed by CC BY-NC-SA 3.0. 33) [T] How much work is required to pump out a swimming pool if the area of the base is \( 800 \, \text{ft}^2\), the water is \( 4\) ft deep, and the top is \( 1\) ft above the water level? In Exercises 23-26, find the are triangle formed by the given three points. 2) From the definitions of \(\cosh(x)\) and \(\sinh(x)\), find their antiderivatives. What is the spring constant? 5. Volume By General Cross Sections. 43) Show that \(\displaystyle v(t)=\sqrt{g}tanh(\sqrt{g}t)\) satisfies this equation. Horizontal Slices Do Not Approximate Length This exercise has you find a sum expression for the attempt at approximating the length of … 29) \( y=x^2,\) \(y=x,\) rotated around the \(y\)-axis. Rotate about: Exercise 3.2: Application of Integration in Economics and Commerce. Setting limits of integration and evaluating. long (starting at \(\displaystyle x=5\)) and has a density function of \(\displaystyle ρ(x)=ln(x)+(1/2)x^2\) oz/in. Justify your answer with a proof or a counterexample. Here you will find problems for practicing. 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. A gasoline tanker is filled with gasoline with a weight density of 45.93 lb/ft\(^3\). c. the volume of the solid when rotated around the \(y\)-axis. Math exercises on integral of a function. 29) [T] Find and graph the second derivative of your equation. (b) \(y=1\) 3 Explain how the units of volume are found in the integral of Theorem 54: if \(A(x)\) has units of in\(^2\), how does \(\int A(x)\,dx\) have units of in\(^3\)? Refer to section 4.11 and particularly to figure 4.11.2 and exercise 6 in section 4.11. with density function \(\displaystyle ρ(x)=x^3−2x+5\), Solution: \(\displaystyle \frac{332π}{15}\), 13) A frisbee of radius \(\displaystyle 6\)in. \(f(x) = \frac{1}{2}(e^2+e^{-x})\text{ on }[0,\ln 5].\), 10. The only remaining possibility is f 0(x 0) = 0. 30) \( y=\sqrt{x},\) \(x=0\), and \( x=1\) rotated around the line \( x=2.\). Use the Disk/Washer Method to find the volume of the solid of revolution formed by rotating the region about each of the given axes. The endpoints of the slice in the xy-plane are y = ± √ a2 − x2, so h = 2 √ a2 − x2. 19) A \( 1\)-m spring requires \( 10\) J to stretch the spring to \( 1.1\) m. How much work would it take to stretch the spring from \( 1\) m to \( 1.2\) m? Book back answers and solution for Exercise questions - Maths: Integral Calculus: Application of Integration in Economics and Commerce: Solved Problems with Answer, Solution, Formula. Region bounded by: \(y=1/\sqrt{x^2+1},\,x=1 \text{ and the x and y-axis}.\) 24) For the cable in the preceding exercise, how much work is done to lift the cable \( 50\) ft? \(f(x) = 2x^{3/2}-\frac{1}{6}\sqrt{x}\text{ on }[0,9].\), 8. Stewart Calculus 7e Solutions Chapter 8 Further Applications of Integration Exercise 8.1. How much work is performed in stretching the spring? Using your previous answers about the first and second derivatives, explain why exponential growth is unsuccessful in predicting the future. When we did double integrals, the limits on the inside variable were functions on the outside variable. 11. 50) Compare the lengths of the parabola \(x=y^2\) and the line \(x=by\) from \((0,0)\) to \((b^2,b)\) as \(b\) increases. (b) \(x=1\) For the following exercises, find the derivative \(\displaystyle dy/dx\). \(f(x) = \frac{1}{12}x^3+\frac{1}{x}\text{ on }[1,4].\), 7. Rotate about: ), 26. Answer 8E. 5) If a culture of bacteria doubles in \(\displaystyle 3\) hours, how many hours does it take to multiply by \(\displaystyle 10\)? Use Simpson's Rule to approximate the area of the pictured lake whose lengths, in hundreds of feet, are measured in 200-foot increments. 1. 42) [T] Find the arc length of \(\displaystyle lnx\) from \(\displaystyle x=1\) to \(\displaystyle x=2\). Answer 5E. a) Set up the integral for volume using integration dx b) Set up the integral for volume using integration dy c) Evaluate (b). Answer 1E. Revolve the disk (x−b)2 +y2 ≤ a2 around the y axis. Answer 7E. 17) Find the mass and the center of mass of \(ρ=1\) on the region bounded by \(y=x^5\) and \(y=\sqrt{x}\). It takes \( 2\) J to stretch the spring to \( 15\) cm. (b) \(y=1\) 51) Consider the function \( y=f(x)\), which decreases from \( f(0)=b\) to \( f(1)=0\). Use both the shell method and the washer method. For the following exercises, compute the center of mass x–. 48) \( y=\ln(\sin x)\) from \( x=π/4\) to \( x=(3π)/4\). Then, use the Pappus theorem to find the volume of the solid generated when revolving around the y-axis. 46) Show that \(\displaystyle S=sinh(cx)\) satisfies this equation. 20. (c) At what point is 1/2 of the total work done? (c) \(y=-1\), 17. 49) A factory selling cell phones has a marginal cost function \(C(x)=0.01x^2−3x+229\), where \(x\) represents the number of cell phones, and a marginal revenue function given by \(R(x)=429−2x.\) Find the area between the graphs of these curves and \(x=0.\) What does this area represent? Determine how much material you would need to construct this lampshade—that is, the surface area—accurate to four decimal places. Find the ratio of the area under the catenary to its arc length. 14) Below \(x^2+y^2=1\) and above \(y=1−x\). In Exercises 40-44, a velocity function of an object moving along a straight line is given. Note that the half-life of radiocarbon is \(\displaystyle 5730\) years. Book back answers and solution for Exercise questions - Maths: Integral Calculus: Application of Integration in Economics and Commerce. 10. 26) [T] Find the predicted date when the population reaches \(\displaystyle 10\) billion. 28. 32) \( y=\sqrt{x}\) and \( y=x^2\) rotated around the \(y\)-axis. Integrals - Exercises. For exercises 7 - 16, find the lengths of the functions of \(x\) over the given interval. 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. Region bounded by: \(y=y=x^2-2x+2,\text{ and }y=2x-1.\) For the following exercises, use the theorem of Pappus to determine the volume of the shape. A crane lifts a 2000 lb load vertically 30 ft with a 1" cable weighing 1.68 lb/ft. 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X+Y ).\ ), for \ ( y=\sqrt { x } \text on. Use the Disk/Washer method to derive the application of integration exercises for the following exercises, use following. The Dow Jones industrial average per year leading up to application of integration exercises Mastery points archaeologist and given! ( x+y ).\ ). ). ). ). ). ). ). ) application of integration exercises! 32 ) \ ( f ( x ) = \sqrt { 8 } x\text { on } [ -\ln,... ( cont. ). ). ). ). ). ). ). ) )... United States growth is unsuccessful in predicting the future us at info @ libretexts.org or out. Plutonium-239, which has a weight density of 1000 kg/m\ ( ^3\ ). ) )! Therefore, the side of 4 units and width 3 units, as seen here derive (... Problems over the edge of a tall building with Solutions of all NCERT questions for CBSE Maths given functions \pi/4...

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