9.E: Applications of Integration (Exercises)
- Page ID
- 3094
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)These are homework exercises to accompany David Guichard's "General Calculus" Textmap. Complementary General calculus exercises can be found for other Textmaps and can be accessed here.
9.1: Area Between Curves
Find the area bounded by the curves.
Ex 9.1.1$\ds y=x^4-x^2$ and $\ds y=x^2$ (the part to the right of the $y$-axis) (answer)
Ex 9.1.2$\ds x=y^3$ and $\ds x=y^2$ (answer)
Ex 9.1.3$\ds x=1-y^2$ and $y=-x-1$ (answer)
Ex 9.1.4$\ds x=3y-y^2$ and $x+y=3$ (answer)
Ex 9.1.5$y=\cos(\pi x/2)$ and $\ds y=1- x^2$ (in the first quadrant) (answer)
Ex 9.1.6$y=\sin(\pi x/3)$ and $y=x$ (in the first quadrant) (answer)
Ex 9.1.7$\ds y=\sqrt{x}$ and $\ds y=x^2$ (answer)
Ex 9.1.8$\ds y=\sqrt x$ and $\ds y=\sqrt{x+1}$, $0\le x\le 4$ (answer)
Ex 9.1.9$x=0$ and $\ds x=25-y^2$ (answer)
Ex 9.1.10$y=\sin x\cos x$ and $y=\sin x$, $0\le x\le \pi$ (answer)
Ex 9.1.11$\ds y=x^{3/2}$ and $\ds y=x^{2/3}$ (answer)
Ex 9.1.12$\ds y=x^2-2x$ and $y=x-2$ (answer)
The following three exercises expand on the geometric interpretation of the hyperbolic functions. Refer to section 4.11and particularly to figure 4.11.2 and exercise 6in section 4.11.
Ex 9.1.13Compute $\ds \int \sqrt{x^2 -1}\,dx $ using the substitution $u=\arccosh x$, or $x=\cosh u$; use exercise 6in section 4.11.
Ex 9.1.14Fix $t>0$. Sketch the region $R$ in the right half plane bounded by the curves $y=x\tanh t$, $y=-x\tanh t$, and $\ds x^2-y^2 =1$. Note well: $t$ is fixed, the plane is the $x$-$y$ plane.
Ex 9.1.15Prove that the area of $R$ is $t$.
9.2: Distance, Velocity, and Acceleration
For each velocity function find both the net distance and the total distance traveled during the indicated time interval (graph $v(t)$ to determine when it's positive and when it's negative):
Ex 9.2.1$v=\cos(\pi t)$, $0\le t\le 2.5$ (answer)
Ex 9.2.2$v=-9.8t+49$, $0\le t\le 10$ (answer)
Ex 9.2.3$v=3(t-3)(t-1)$, $0\le t\le 5$ (answer)
Ex 9.2.4$v=\sin(\pi t/3)-t$, $0\le t\le 1$ (answer)
Ex 9.2.5An object is shot upwards from ground level with an initial velocity of 2 meters per second; it is subject only to the force of gravity (no air resistance). Find its maximum altitude and the time at which it hits the ground. (answer)
Ex 9.2.6An object is shot upwards from ground level with an initial velocity of 3 meters per second; it is subject only to the force of gravity (no air resistance). Find its maximum altitude and the time at which it hits the ground. (answer)
Ex 9.2.7An object is shot upwards from ground level with an initial velocity of 100 meters per second; it is subject only to the force of gravity (no air resistance). Find its maximum altitude and the time at which it hits the ground. (answer)
Ex 9.2.8An object moves along a straight line with acceleration given by $a(t) = -\cos(t)$, and $s(0)=1$ and $v(0)=0$. Find the maximum distance the object travels from zero, and find its maximum speed. Describe the motion of the object. (answer)
Ex 9.2.9An object moves along a straight line with acceleration given by $a(t) = \sin(\pi t)$. Assume that when $t=0$, $s(t)=v(t)=0$. Find $s(t)$, $v(t)$, and the maximum speed of the object. Describe the motion of the object. (answer)
Ex 9.2.10An object moves along a straight line with acceleration given by $a(t) = 1+\sin(\pi t)$. Assume that when $t=0$, $s(t)=v(t)=0$. Find $s(t)$ and $v(t)$. (answer)
Ex 9.2.11An object moves along a straight line with acceleration given by $a(t) = 1-\sin(\pi t)$. Assume that when $t=0$, $s(t)=v(t)=0$. Find $s(t)$ and $v(t)$. (answer)
9.3: Volume
9.4: Average Value of a Function
Ex 9.4.1Find the average height of $\cos x$ over the intervals $[0,\pi/2]$, $[-\pi/2,\pi/2]$, and $[0,2\pi]$. (answer)
Ex 9.4.2Find the average height of $\ds x^2$ over the interval $[-2,2]$. (answer)
Ex 9.4.3Find the average height of $\ds 1/x^2$ over the interval $[1,A]$. (answer)
Ex 9.4.4Find the average height of $\ds \sqrt{1-x^2}$ over the interval $[-1,1]$. (answer)
Ex 9.4.5An object moves with velocity $\ds v(t)=-t^2+1$ feet per second between $t=0$ and $t=2$. Find the average velocity and the average speed of the object between $t=0$ and $t=2$. (answer)
Ex 9.4.6The observation deck on the 102nd floor of the Empire State Building is 1,224 feet above the ground. If a steel ball is dropped from the observation deck its velocity at time $t$ is approximately $v(t)=-32t$ feet per second. Find the average speed between the time it is dropped and the time it hits the ground, and find its speed when it hits the ground. (answer)
9.5: Work
Ex 9.5.1 How much work is done in lifting a 100 kilogram weight from the surface of the earth to an orbit 35,786 kilometers above the surface of the earth? (answer)
Ex 9.5.2 How much work is done in lifting a 100 kilogram weight from an orbit 1000 kilometers above the surface of the earth to an orbit 35,786 kilometers above the surface of the earth? (answer)
Ex 9.5.3 A water tank has the shape of an upright cylinder with radius $r=1$ meter and height 10 meters. If the depth of the water is 5 meters, how much work is required to pump all the water out the top of the tank? (answer)
Ex 9.5.4 Suppose the tank of the previous problem is lying on its side, so that the circular ends are vertical, and that it has the same amount of water as before. How much work is required to pump the water out the top of the tank (which is now 2 meters above the bottom of the tank)? (answer)
Ex 9.5.5 A water tank has the shape of the bottom half of a sphere with radius $r=1$ meter. If the tank is full, how much work is required to pump all the water out the top of the tank? (answer)
Ex 9.5.6 A spring has constant $k=10$ kg/s$^2$. How much work is done in compressing it $1/10$ meter from its natural length? (answer)
Ex 9.5.7 A force of 2 Newtons will compress a spring from 1 meter (its natural length) to 0.8 meters. How much work is required to stretch the spring from 1.1 meters to 1.5 meters? (answer)
Ex 9.5.8 A 20 meter long steel cable has density 2 kilograms per meter, and is hanging straight down. How much work is required to lift the entire cable to the height of its top end? (answer)
Ex 9.5.9 The cable in the previous problem has a 100 kilogram bucket of concrete attached to its lower end. How much work is required to lift the entire cable and bucket to the height of its top end? (answer)
Ex 9.5.10 Consider again the cable and bucket of the previous problem. How much work is required to lift the bucket 10 meters by raising the cable 10 meters? (The top half of the cable ends up at the height of the top end of the cable, while the bottom half of the cable is lifted 10 meters.) (answer)
9.6: Center of Mass
Ex 9.6.1 A beam 10 meters long has density \( \sigma(x)=x^2\) at distance \(x\) from the left end of the beam. Find the center of mass \( \bar x\). (answer)
Ex 9.6.2 A beam 10 meters long has density \(\sigma(x)=\sin(\pi x/10)\) at distance \(x\) from the left end of the beam. Find the center of mass \( \bar x\). (answer)
Ex 9.6.3 A beam 4 meters long has density \( \sigma(x)=x^3\) at distance \(x\) from the left end of the beam. Find the center of mass \( \bar x\). (answer)
Ex 9.6.4 Verify that \(\ds\int 2x\arccos x\,dx= x^2\arccos x-{x\sqrt{1-x^2}\over2}+{\arcsin x\over 2}+C\).
Ex 9.6.5 A thin plate lies in the region between \( y=x^2\) and the \(x\)-axis between \(x=1\) and \(x=2\). Find the centroid. (answer)
Ex 9.6.6 A thin plate fills the upper half of the unit circle \( x^2+y^2=1\). Find the centroid. (answer)
Ex 9.6.7 A thin plate lies in the region contained by \(y=x\) and \( y=x^2\). Find the centroid. (answer)
Ex 9.6.8 A thin plate lies in the region contained by \( y=4-x^2\) and the \(x\)-axis. Find the centroid. (answer)
Ex 9.6.9 A thin plate lies in the region contained by \( y=x^{1/3}\) and the \(x\)-axis between \(x=0\) and \(x=1\). Find the centroid. (answer)
Ex 9.6.10 A thin plate lies in the region contained by \( \sqrt{x}+\sqrt{y}=1\) and the axes in the first quadrant. Find the centroid. (answer)
Ex 9.6.11 A thin plate lies in the region between the circle \( x^2+y^2=4\) and the circle \( x^2+y^2=1\), above the \(x\)-axis. Find the centroid. (answer)
Ex 9.6.12 A thin plate lies in the region between the circle \( x^2+y^2=4\) and the circle \( x^2+y^2=1\) in the first quadrant. Find the centroid. (answer)
Ex 9.6.13 A thin plate lies in the region between the circle \( x^2+y^2=25\) and the circle \( x^2+y^2=16\) above the \(x\)-axis. Find the centroid. (answer)
9.7: Kinetic energy and Improper Integrals
Ex 9.7.1 Is the area under $y=1/x$ from 1 to infinity finite or infinite? If finite, compute the area. (answer)
Ex 9.7.2 Is the area under $\ds y=1/x^3$ from 1 to infinity finite or infinite? If finite, compute the area. (answer)
Ex 9.7.3 Does $\ds\int_0^\infty x^2+2x-1\,dx$ converge or diverge? If it converges, find the value. (answer)
Ex 9.7.4 Does $\ds\int_1^\infty 1/\sqrt{x}\,dx$ converge or diverge? If it converges, find the value. (answer)
Ex 9.7.5 Does $\ds\int_0^\infty e^{-x }\,dx$ converge or diverge? If it converges, find the value. (answer)
Ex 9.7.6 $\ds\int_0^{1/2} (2x-1)^{-3}\,dx$ is an improper integral of a slightly different sort. Express it as a limit and determine whether it converges or diverges; if it converges, find the value. (answer)
Ex 9.7.7 Does $\ds\int_0^1 1/\sqrt{x}\,dx$ converge or diverge? If it converges, find the value. (answer)
Ex 9.7.8 Does $\ds\int_0^{\pi/2} \sec^2x\,dx$ converge or diverge? If it converges, find the value. (answer)
Ex 9.7.9 Does $\ds\int_{-\infty}^\infty{x^2\over 4+x^6}\,dx$ converge or diverge? If it converges, find the value. (answer)
Ex 9.7.10 Does $\ds\int_{-\infty}^\infty x\,dx$ converge or diverge? If it converges, find the value. Also find the Cauchy Principal Value, if it exists. (answer)
Ex 9.7.11 Does $\ds\int_{-\infty}^\infty \sin x\,dx$ converge or diverge? If it converges, find the value. Also find the Cauchy Principal Value, if it exists. (answer)
Ex 9.7.12 Does $\ds\int_{-\infty}^\infty \cos x\,dx$ converge or diverge? If it converges, find the value. Also find the Cauchy Principal Value, if it exists. (answer)
Ex 9.7.13 Suppose the curve $y=1/x$ is rotated around the $x$-axis generating a sort of funnel or horn shape, called Gabriel's horn or Toricelli's trumpet. Is the volume of this funnel from $x=1$ to infinity finite or infinite? If finite, compute the volume. (answer)
Ex 9.7.14 An officially sanctioned baseball must be between 142 and 149 grams. How much work, in Newton-meters, does it take to throw a ball at 80 miles per hour? At 90 mph? At 100.9 mph? (According to the Guinness Book of World Records, at \url{www.baseball-almanac.com/recb...rb_guin.shtml} {\vb|www.baseball-almanac.com/recb...shtml|}\endurl, "The greatest reliably recorded speed at which a baseball has been pitched is 100.9 mph by Lynn Nolan Ryan (California Angels) at Anaheim Stadium in California on August 20, 1974.'') (answer)
9.8: Probability
Ex 9.8.1Verify that $\ds \int_1^\infty e^{-x/2}\,dx=2/\sqrt{e}$.
Ex 9.8.2Show that the function in example 9.8.5 is a probability density function. Compute the mean and standard deviation. (answer)
Ex 9.8.3Compute the mean and standard deviation of the uniform distribution on $[a,b]$. (See example 9.8.3.) (answer)
Ex 9.8.4What is the expected value of one roll of a fair six-sided die? (answer)
Ex 9.8.5What is the expected sum of one roll of three fair six-sided dice? (answer)
Ex 9.8.6Let $\mu$ and $\sigma$ be real numbers with $\sigma> 0$. Show that $$N(x) = {1\over\sqrt{2\pi} \sigma} e^{-{(x-\mu)^2\over 2\sigma^2}}$$ is a probability density function. You will not be able to compute this integral directly; use a substitution to convert the integral into the one from example 9.8.4. The function $N$ is the probability density function of thenormal distribution with mean $\mu$ and standard deviation $\sigma$. Show that the mean of the normal distribution is $\mu$ and the standard deviation is $\sigma$.
Ex 9.8.7Let $$ f(x) = \cases{ \ds{1\over x^2 } & $|x| \geq 1$\cr 0 & $|x| < 1$\cr} $$ Show that $f$ is a probability density function, and that the distribution has no mean.
Ex 9.8.8Let $$ f(x) = \cases{ x & $-1\leq x \leq 1$\cr 1 & $1 < x \leq 2$\cr 0 & otherwise.\cr} $$ Show that $\ds \int_{-\infty }^\infty f(x)\,dx = 1$. Is $f$ a probability density function? Justify your answer.
Ex 9.8.9If you have access to appropriate software, find $r$ so that $$ \int_{-\infty}^{10+r} f(x)\,dx + \int_{10+r}^{\infty} f(x)\,dx \approx0.05. $$ Discuss the impact of using this new value of $r$ to decide whether to investigate the chip manufacturing process. (answer)
9.9: Arc Length
Ex 9.9.1 Find the arc length of \( f(x)=x^{3/2}\) on \([0,2]\). (answer)
Ex 9.9.2 Find the arc length of \( f(x) = x^2/8-\ln x\) on \([1,2]\). (answer)
Ex 9.9.3 Find the arc length of \( f(x) = (1/3)(x^2 +2)^{3/2}\) on the interval \([0,a]\). (answer)
Ex 9.9.4 Find the arc length of \(f(x)=\ln(\sin x)\) on the interval \([\pi/4,\pi/3]\). (answer)
Ex 9.9.5 Let \(a>0\). Show that the length of \(y=\cosh x\) on \([0,a]\) is equal to \( \int _0 ^a \cosh x\,dx\).
Ex 9.9.6 Find the arc length of \(f(x)=\cosh x\) on \([0, \ln 2]\). (answer)
Ex 9.9.7 Set up the integral to find the arc length of \(\sin x\) on the interval \([0,\pi]\); do not evaluate the integral. If you have access to appropriate software, approximate the value of the integral. (answer)
Ex 9.9.8 Set up the integral to find the arc length of \( y=xe^{-x}\) on the interval \([2,3]\); do not evaluate the integral. If you have access to appropriate software, approximate the value of the integral. (answer)
Ex 9.9.9 Find the arc length of \( y=e^x\) on the interval \([0,1]\). (This can be done exactly; it is a bit tricky and a bit long.) (answer)
9.10: Surface Area
Ex 9.10.1Compute the area of the surface formed when $\ds f(x)=2\sqrt{1-x}$ between $-1$ and $0$ is rotated around the $x$-axis. (answer)
Ex 9.10.2Compute the surface area of example 9.10.2 by rotating $\ds f(x)=\sqrt x$ around the $x$-axis.
Ex 9.10.3Compute the area of the surface formed when $\ds f(x)=x^3$ between $1$ and $3$ is rotated around the $x$-axis. (answer)
Ex 9.10.4Compute the area of the surface formed when $\ds f(x)=2 +\cosh (x)$ between $0$ and $1$ is rotated around the $x$-axis. (answer)
Ex 9.10.5Consider the surface obtained by rotating the graph of $\ds f(x)=1/x$, $x\geq 1$, around the $x$-axis. This surface is calledGabriel's horn or Toricelli's trumpet. In exercise 13 in section 9.7 we saw that Gabriel's horn has finite volume. Show that Gabriel's horn has infinite surface area.
Ex 9.10.6Consider the circle $\ds (x-2)^2+y^2 = 1$. Sketch the surface obtained by rotating this circle about the $y$-axis. (The surface is called a torus.) What is the surface area? (answer)
Ex 9.10.7Consider the ellipse with equation $\ds x^2/4+y^2 = 1$. If the ellipse is rotated around the $x$-axis it forms an ellipsoid. Compute the surface area. (answer)
Ex 9.10.8Generalize the preceding result: rotate the ellipse given by $\ds x^2/a^2+y^2/b^2=1$ about the $x$-axis and find the surface area of the resulting ellipsoid. You should consider two cases, when $a>b$ and when $a < b$. Compare to the area of a sphere. (answer)