application of integration exercises

A catenoid in nature can be found when stretching soap between two rings. Have questions or comments? 51) The base is the region between $ y=x$ and $ y=x^2$. 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. Find area of the shaded region below. How much work is performed in compressing the spring? Rotate about: In Exercises 13-20, set up the integral to compute the arc length of the function on the given interval. For exercises 26 - 37, graph the equations and shade the area of the region between the curves. 6) If bacteria increase by a factor of $\displaystyle 10$ in $\displaystyle 10$ hours, how many hours does it take to increase by $\displaystyle 100$? 13) If $\displaystyle y=1000$ at $\displaystyle t=3$ and $\displaystyle y=3000$ at $\displaystyle t=4$, what was $\displaystyle y_0$ at $\displaystyle t=0$? Answer 2E. (Hint: Use the theorem of Pappus.). Use symmetry to help locate the center of mass whenever possible. For exercises 7 - 13, graph the equations and shade the area of the region between the curves. 19) You are cooling a turkey that was taken out of the oven with an internal temperature of $\displaystyle 165°F$. (Hint: Since $ f(x)$ is one-to-one, there exists an inverse $ f^{−1}(y)$.). 1. A force of 50 lb compresses a spring from a natural length of 18 in to 12 in. Volume By General Cross Sections. 28) $ y=x^3$, $y=0$, $x=0$, and $ y=8$ rotated around the $y$-axis. Starting from $\displaystyle 8$ million (New York) and $\displaystyle 6$ million (Los Angeles), when are the populations equal? Math exercises on integral of a function. If the race is over in 1 hour, who won the race and by how much? 14) If $\displaystyle y=100$ at $\displaystyle t=4$ and $\displaystyle y=10$ at $\displaystyle t=8$, when does $\displaystyle y=1$? Source: http:/www.sfgenealogy.com/sf/history/hgpop.htm. Use both the shell method and the washer method. Learn Chapter 8 Application of Integrals (AOI) of Class 12 free with solutions of all NCERT Questions for CBSE Maths. Stewart Calculus 7e Solutions Pdf. 44) A light bulb is a sphere with radius $1/2$ in. This page contains a list of commonly used integration formulas with examples,solutions and exercises. Volumes by shells 4C-1 Assume that 0 < a < b. Where is it increasing and what is the meaning of this increase? We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. In primary school, we learned how to find areas of shapes with straight sides (e.g. Use a graphing calculator to graph the data and the exponential curve together. A skew right circular cone with height of 10 and base radius of 5. Exercise 3.3 . Answer 8E. 31. 46) Show that $\displaystyle S=sinh(cx)$ satisfies this equation. 13. Then, find the volume when the region is rotated around the $y$-axis. Stewart Calculus 7e Solutions Chapter 5 Applications of Integration Exercise 5.1 . A similar argument deals with the case when f 0(x 0) < 0. 17) Find the mass and the center of mass of $ρ=1$ on the region bounded by $y=x^5$ and $y=\sqrt{x}$. The functions $f(x)=\cos (2x)\text{ and }g(x) =\sin x$ intersect infinitely many times, forming an infinite number of repeated, enclosed regions. 23. Determine its area by integrating over the $y$-axis. Using your previous answers about the first and second derivatives, explain why exponential growth is unsuccessful in predicting the future. 17) A $ 12$-in. Use a calculator to determine intersection points, if necessary, to two decimal places. 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. For exercises 1 - 6, find the volume generated when the region between the two curves is rotated around the given axis. 7. A 5 m tall cylindrical tank with radius of 2 m is filled with 3 m of gasoline, with a mass density of 737.22 kg/m$^3$. Answer 7E. 30) A general cylinder created by rotating a rectangle with vertices $\displaystyle (0,0), (a,0),(0,b),$ and $\displaystyle (a,b)$ around the $\displaystyle y$ -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? (Note: $f'(x)$ is not defined at the endpoints.). Integrals - Exercises. Then, use the washer method to find the volume when the region is revolved around the $y$-axis. Watch the recordings here on Youtube! For the following exercises, compute the center of mass x–. 41) [T] $x=\sqrt{4−y^2}$ and $ y^2=1+x^2$, 43) [T] $y=\sin^3x+2,\quad y=\tan x,\quad x=−1.5,$ and $x=1.5$, 44) [T] $y=\sqrt{1−x^2}$ and $y^2=x^2$, 45) [T] $y=\sqrt{1−x^2}$ and $y=x^2+2x+1$, 47) [T] $y=\cos x,\quad y=e^x,\quad x=−π,\quad$ and$\quad x=0$. (c) the x-axis Solution: $\displaystyle \frac{1}{2}sinh(2x+1)+C$, Solution: $\displaystyle \frac{1}{2}sinh^2(x^2)+C$, Solution: $\displaystyle \frac{1}{3}cosh^3(x)+C$, 25) $\displaystyle \frac{sinh(x)}{1+cosh(x)}$, Solution: $\displaystyle ln(1+cosh(x))+C$, Solution: $\displaystyle cosh(x)+sinh(x)+C$, 28) $\displaystyle (cosh(x)+sinh(x))^n$. For the following exercises, solve each problem. For the following exercises, find the derivative $\displaystyle dy/dx$. Find out how much rope you need to buy, rounded to the nearest foot. Sand leaks from the bag at a rate of 1/4 lb/s. Region bounded by: $y=4-x^2\text{ and }y=0.$ Rotate about: 13. True or False? 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. $f(x) = \ln \left ( \sin x \right ) \text{ on }[\pi/6,\pi/2].$, 12. 10) A cone of radius $ r$ and height $ h$ has a smaller cone of radius $ r/2$ and height $ h/2$ removed from the top, as seen here. Use a graphing calculator to graph the data and the exponential curve together. A right triangle cone with height of 10 and whose base is a right, isosceles triangle with side length 4. For exercises 5 - 8, use the requested method to determine the volume of the solid. 30. What is the spring constant? You know the cooling constant is $k=0.00824$ °F/min. 20) You are trying to thaw some vegetables that are at a temperature of $\displaystyle 1°F$. 1. 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. 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. 25) [T] Find and graph the second derivative of your equation. For exercises 41 - 45, draw the region bounded by the curves. Exercise 3.3: Application of Integration in Economics and Commerce. Answer 7E. 49) $\displaystyle \frac{d}{dx}ln(x+\sqrt{x^2+1})=\frac{1}{\sqrt{1+x^2}}$, 50) $\displaystyle \frac{d}{dx}ln(\frac{x−a}{x+a})=\frac{2a}{(x^2−a^2)}$, 51) $\displaystyle \frac{d}{dx}ln(\frac{1+\sqrt{1−x^2}}{x})=−\frac{1}{x\sqrt{1−x^2}}$, 52) $\displaystyle \frac{d}{dx}ln(x+\sqrt{x^2−a^2})=\frac{1}{\sqrt{x^2−a^2}}$, 53) $\displaystyle ∫\frac{dx}{xln(x)ln(lnx)}=ln(ln(lnx))+C$. Slices perpendicular to the $x$-axis are semicircles. 45) [T] A lampshade is constructed by rotating $ y=1/x$ around the $x$-axis from $ y=1$ to $ y=2$, as seen here. Determine which method you think would be easiest to use to calculate the volume generated when the function is rotated around the specified axis. There are also some electronics applications in this section.. Introduction Exercise 3 on Applications of Integration will focus on Kinematic Problems. A 50 m rope, with a mass density of 0.2 kg/m, hangs over the edge of a tall building. Compute the total work performed in pumping all the gasoline to the top of the tank. These are homework exercises to accompany OpenStax's "Calculus" Textmap. Worksheets 1 to 15 are topics that are taught in MATH108. For the following exercises, use the function $\displaystyle lnx$. Slices perpendicular to the $x$-axis are semicircles. Evaluate the triple integral with order dz dy dx. Does your answer agree with the volume of a cone? Rotate about: 43) [T] $ x=\sin(πy^2)$ and $ x=\sqrt{2}y$ rotated around the $x$-axis. 194 Chapter 9 Applications of Integration 11. y = x3/2 and 2/3 ⇒ 12. y = x2 −2and ⇒ The following three exercises expand on the geometric interpretation of the hyperbolic functions. Region bounded by: $y=\sqrt{x},\,y=0\text{ and }x=1.$ This doughnut shape is known as a torus. Where is it increasing? (d) $y=2$, 16. Topic 6 Application of Integration 6.1 Volumes Exercise 6.1 Find the volume of the solid obtained by rotating the region enclosed by the given curves \[y=x^2, \quad y=x\] about $y$ -axis. 22. Textbook Authors: Stewart, James , ISBN-10: 1285740629, ISBN-13: 978-1-28574-062-1, Publisher: Cengage Stewart Calculus 7e Solutions Chapter 5 Applications of Integration Exercise 5.1. A right circular cone with height of 10 and base radius of 5. 0 b= 4πa2b/3. After you have run 4 seconds the raptor is 32 meters from the corner. 2. Find the area of the region bounded by the curve y^2 =x and the lines x = 1 , x = 4 and the x axis . Answer 10E. The aim here is to illustrate that integrals (deﬁnite integrals) have applications to practical things. 37) $\displaystyle ∫\frac{dx}{a^2−x^2}$, Solution: $\displaystyle \frac{1}{a}tanh^{−1}(\frac{x}{a})+C$, 38) $\displaystyle ∫\frac{dx}{\sqrt{x^2+1}}$, 39) $\displaystyle ∫\frac{xdx}{\sqrt{x^2+1}}$, Solution: $\displaystyle \sqrt{x^2+1}+C$, 40) $\displaystyle ∫−\frac{dx}{x\sqrt{1−x^2}}$, 41) $\displaystyle ∫\frac{e^x}{\sqrt{e^{2x}−1}}$, Solution: $\displaystyle cosh^{−1}(e^x)+C$. If $\displaystyle 1$ barrel containing $\displaystyle 10kg$ of plutonium-239 is sealed, how many years must pass until only $\displaystyle 10g$ of plutonium-239 is left? 19) The length of $y$ for $x=3−\sqrt{y}$ from $y=0$ to $y=4$. A gasoline tanker is filled with gasoline with a weight density of 45.93 lb/ft$^3$. 49) A better approximation of the volume of a football is given by the solid that comes from rotating $ y=\sin x) around the \(x$-axis from $ x=0$ to $ x=π$. Unless otherwise noted, LibreTexts content is licensed by CC BY-NC-SA 3.0. 48) Use the method of shells to find the volume of a cylinder with radius $ r$ and height $ h$. By the end of this lesson you should be able to: • use the definite integral to find areas above and below the axis. 9. 2. 9) A pyramid with height 5 units, and an isosceles triangular base with lengths of 6 units and 8 units, as seen here. 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. What do you notice? Rotate about: 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}$. $f(x) = \sec x\text{ on }[-\pi/4, \pi/4]$. 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: $\displaystyle −\frac{1}{x^2}$. E. 18.01 EXERCISES 4C. 2. 52) The tortoise versus the hare: The speed of the hare is given by the sinusoidal function $H(t)=(1/2)−(1/2)\cos(2πt)$ whereas the speed of the tortoise is $T(t)=\sqrt{t}$, where $t$ is time measured in hours and speed is measured in kilometers per hour. 42) [T] $ y=3x^3−2,y=x$, and $ x=2$ rotated around the $y$-axis. $f(x) = \sqrt{1-x^2}\text{ on }[-1,1].$ (Note: $f'(x)$ is not defined at the endpoints. 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. A force of 20 lb stretches a spring from a natural length of 6 in to 8 in. 5) If a culture of bacteria doubles in $\displaystyle 3$ hours, how many hours does it take to multiply by $\displaystyle 10$? 4. 50) An amusement park has a marginal cost function $C(x)=1000e−x+5$, where $x$ represents the number of tickets sold, and a marginal revenue function given by $R(x)=60−0.1x$. Answer 9E. 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. In your own words, explain how the Disk and Washer Methods are related. 8) A tetrahedron with a base side of 4 units,as seen here. (a) the x-axis The constant $\displaystyle c$ is the ratio of cable density to tension. Setting limits of integration and evaluating. In Exercises 18-22, find the area of the enclosed region in two ways: 44) Derive the previous expression for $\displaystyle v(t)$ by integrating $\displaystyle \frac{dv}{g−v^2}=dt$. 13) The base is the region under the parabola $ y=1−x^2$ in the first quadrant. T/F: The are between curves is always positive. #Application of Integration Question#1 Exercise 8.1 Class 12 2. concrete, with a weight density of 150 lb/ft$^3$. 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