Homework 12: Calculus and Analytic Geometry

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au
and

Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA
ram@math.wisc.edu

Last updates: 11 February 2010

Problem A. Length of a plane curve.

1. Use integration to show that the circumference of a circle of radius $r$ is $2\pi r$
2. Find the length of the curve $y=x^{\frac{3}{2}}$ between $x=-1$ and $x=8$
3. Find the total length of the curve determine by the equations $x=a{\cos }^3\theta$ and $y=a{\sin }^3\theta$
4. Find the length of the curve $y=\frac{1}{3}{\left(x^2+2\right)}^{\frac{3}{2}}$ from $x=0$ to $x=3$
5. Find the length of the curve $y=x^{\frac{3}{2}}$ from (0,0) to (4,8)
6. Find the length of the curve $9x^2=4y^3$ from (0,0) to $\left(2\times 3^{\frac{1}{2}},3\right)$
7. Find the length of the curve $y=\frac{1}{3}{x}^3+\frac{1}{4}x$ from $x=1$ to $x=3$
8. Find the length of the curve $x=\frac{y^4}{4}+\frac{y^2}{8}$ from $y=1$ to $y=2$
9. Find the length of the curve ${\left(y+1\right)}^2=4x^3$ from $x=0$ to $x=1$
10. Find the distance travelled by a particle $P\left(x,y\right)$ between $t=0$ and $t=\frac{\pi }{2}$ whose position at time $t$ is given by $x=a\cos t+at\sin t,y=a\sin t-at\cos t,$ where $a$ is a positive constant
11. Find the length of the curve $x=t-\sin t,y=1-\cos t,0\le t\le \frac{\pi }{2}$
12. Find the distance travelled by the particle $P\left(x,y\right)$ between $t=0$ and $t=4$ if the position at time $t$ is given by $x=\frac{t^2}{2},y=\frac{1}{3}{\left(2t+3\right)}^{\frac{3}{2}}$
13. The position of the particle $P\left(x,y\right)$ at time $t$ is given by $x=\frac{1}{3}{\left(2t+3\right)}^{\frac{3}{2}},y=\frac{t^2}{2}+t.$ Find the distance travelled between $t=0$ and $t=3$
14. Find the length of the curve $x=\frac{3}{5}{y}^{\frac{5}{3}}-\frac{3}{4}{y}^{\frac{1}{3}}$ from $y=0$ to $y=1$
15. Find the length of the curve $y=\frac{2}{3}{x}^{\frac{3}{2}}-\frac{1}{2}{x}^{\frac{1}{2}}$ from $x=0$ to $x=4$
16. Consider the curve $y=f\left(x\right),x\ge 0,$ such that $f\left(0\right)=a.$ Let $s\left(x\right)$ denote the arc length along the curve from $\left(0,a\right)$ to $\left(x,f\left(x\right)\right).$ Find $f\left(x\right)$ if $s\left(x\right)=Ax.$ What are the permissible values of $A$?
17. Consider the curve $y=f\left(x\right),x\ge 0,$ such that $f\left(0\right)=a.$ Let $s\left(x\right)$ denote the arc length along the curve from $\left(0,a\right)$ to $\left(x,f\left(x\right)\right).$ Is it possible for $s\left(x\right)$ to equal $x^n$ with $n>1$? Give a reason for your answer

Problem B. Surface Area

1. Use integration to show that the surface area of a sphere of radius $r$ is $4\pi r^2$
2. Find the surface area of the bagel obtained by rotating the circle $x^2+y^2=r^2$ round the line $y=-r$
3. Find the surface area of the solid generated by rotating the portion of the curve $y=\frac{1}{3}{\left(x^2+2\right)}^{\frac{3}{2}}$ between $x=0$ and $x=3$ about the $x$-axis
4. Find the area of the surface generated by rotating the arc of the curve $y=x^3$ between $x=0$ and $x=1$ about the $x$-axis
5. Find the area of the surface generated by rotating the arc of the curve $y=x^2$ between (0,0) and (2,4) about the $y$-axis
6. The arc of the curve $y=\frac{x^3}{3}+\frac{x}{4}$ from $x=1$ to $x=3$ is rotated about the line $y=-1$. Find the surface area generated
7. The arc of the curve $x=\frac{y^4}{4}+\frac{y^2}{8}$ from $y=1$ to $y=2$ is rotated about the $x$-axis
8. Find the area of the surface obtained by rotating about the $y$-axis the curve $y=\frac{x^2}{2}+\frac{1}{2},0\le x\le 1$
9. Find the area of the surface generated by rotating the curve determined by $x=a{\cos }^3\theta$ and $y=a{\sin }^3\theta$ about the $x$-axis
10. The curve described by the particle $P\left(x,y\right)$ with position given by $x=t+1,y=\frac{t^2}{2}+t,$ from $t=0$ to $t=4$ is rotated about the $y$-axis. Find the surface area that is generated.
11. The loop of the curve $9x^2=y{\left(3-y\right)}^2$ is rotated about the $x$-axis. Find the surface area generated.
12. Find the surface area generated when the curve $y=\left(\frac{2}{3}\right)x^{\frac{3}{2}}-\left(\frac{1}{2},x^{\frac{1}{2}}\right)$ from $x=0$ to $x=4$ is rotated about the $y-axis.$
13. Find the surface area generated when the curve $x=\left(\frac{3}{5}\right)y^{\frac{5}{3}}-\left(\frac{3}{4}\right)y^{\frac{1}{3}}$ from $y=0$ to $y=1$ is rotated about the line $y=-1$

Problem C. Center of Mass

1. Find the center of mass of a thin homogeneous triangular shaped plate of base $b$ and height $h$
2. A thin homogeneous wire is bent to form a semicircle of radius $r.$ Find its center of mass
3. Find the center of mass of a solid hemisphere of radius $r$ if its density at any point $P$ is proportional to the distance between $P$ and the base of the hemisphere
4. Find the center of mass of a thin homogeneous plate covering the area bounded by the parabola $y=h^2-x^2$ and the $x$-axis.
5. Find the center of mass of a thin homogeneous plate covering the area in the first quadrant of the circle $x^2+y^2=a^2$
6. Find the center of mass of a thin homogeneous plate covering the 'triangular' area in the first quadrant between the circle $x^2+y^2=a^2$ and the lines $y=a,x=a$
7. Find the center of mass of a thin homogeneous plate covering the area between the $x$-axis and the curve $y=\sin x$ between $x=0$ and $x=\pi$
8. Find the center of mass of a thin homogeneous plate covering the area between the $y$-axis and the curve $x=2y-y^2$
9. Find the distance, from the base, of the center of mass of a thin triangular plate of base $b$ and $h$ if its density varies as the square root of the distance from the base.
10. Find the distance, from the base, of the center of mass of a thin triangular plate of base $b$ and $h$ if its density varies as the square of the distance from the base.
11. Find the center of mass of a homogeneous right center cone
12. Find the center of mass of a solid right circular cone if the densty varies as the distance from the base.
13. A thin homogeneous wire is bent to form a semicircle of radius $r$; suppose that the density is $d=k\sin \theta$ where $k$ is a constant. Find the center of mass
14. Find the center of gravity of a solid hemisphere of radius $r$
15. Find the center of gravity of athin hemipherical shell of inner radius $r$ and thickness $t$
16. Find the center of gravity of the area bounded by the $x$-axis and the curve $y=c^2-x^2$
17. Find the center of gravity of the area bounded by the $y$-axis and the curve $x=y-y^3,0\le y\le 1$
18. Find the center of gravity of the area bounded by the curve $y=x^2$ and the line $y=4$
19. Find the center of gravity of the area bounded by the curve $y=x-x^2$ and the line $x+y=0$
20. Find the center of gravity of the are bounded by the curve $x=y^2-y$ and the line $y=x$
21. Find the center of gravity of a solid right circular cone of altitude $h$ and base radius $r$
22. Find the center of gravity of the solid generated by rotating about the $y$-axis the area bounded by the curve $y=x^2$ and the line $y=4$
23. The area bounded by the curve $x=y^2-y$ and the line $y=x$ is rotated about the $x$ axis. Find the center of gravity of the solid thus generated.
24. Find the center of gravity of the very thin right circular conical shell of base radius $r$ and height $h$
25. Find the center of gravity of the surface area generated by rotating about the line $x=-r$ the arc of the circle $x^2+y^2=a^2$ that lies in the first quadrant.
26. Find the moment abut the $x$-axis of the arc of the parabola $y=x^{\frac{1}{2}}$ lying between (0,0) and (4,2)
27. Find the center of gravity of the arc of one quadrant of a circle

Problem D. Average value of a function.

1. Explain how to derive the formula of the average value of a function $f\left(x\right)$ as $x$ ranges from $a$ to $b$
2. Compute the average of the numbers $1,2,\dots ,100$
3. Compute the average of the numbers $9,10,\dots ,243$
4. Compute the average of the numbers $-9,-6,-3,\dots ,243$
5. Compute the average of the numbers $3^0,3^1,\dots ,3^{50}$
6. Explain why the average of the numbers $1,\frac{1}{2},\frac{1}{3},\dots ,\frac{1}{100}$ is more than .04615 but less than .04705
7. Explain why the average of the numbers $1,e^{-1},e^{-2},\dots ,e^{-50}$ is more than .02 but less than .04
8. Show that the average of $1,e^{-1},e^{-2},\dots ,e^{-50}$ is equal to .031639534 (up to 7 decimal places).
9. Explain why the average of the numbers $1,\frac{1}{4},\frac{1}{9},\frac{1}{16},\frac{1}{25},\dots ,\frac{1}{10000}$ is more than .00333433 but less than .0133333
10. Graph $f\left(x\right)=\sin \left(x\right),0\le x\le \frac{\pi }{2}$ and find its average value. Indicate the average value on the graph. Draw a rectangle with base $0\le x\le \frac{\pi }{2}$ and with area equal to the area under the graph of $f\left(x\right)$
11. Graph $f\left(x\right)=\sin \left(x\right),0\le x\le 2\pi$ and find its average value. Indicate the average value on the graph. Draw a rectangle with base $0\le x\le 2\pi$ and with area equal to the area under the graph of $f\left(x\right)$
12. Graph $f\left(x\right)={\sin }^2\left(x\right),0\le x\le \frac{\pi }{2}$ and find its average value. Indicate the average value on the graph. Draw a rectangle with base $0\le x\le \frac{\pi }{2}$ and with area equal to the area under the graph of $f\left(x\right)$
13. Graph $f\left(x\right)={\sin }^2\left(x\right),0\le x\le 2\pi$ and find its average value. Indicate the average value on the graph. Draw a rectangle with base $0\le x\le 2\pi$ and with area equal to the area under the graph of $f\left(x\right)$
14. Graph $f\left(x\right)={\left(2x+1\right)}^{\frac{1}{2}},4\le x\le 12$ and find its average value. Indicate the average value on the graph. Draw a rectangle with base $4\le x\le 12$ and with area equal to the area under the graph of $f\left(x\right)$
15. Graph $f\left(x\right)=\frac{1}{2}+\frac{1}{2}\cos 2x,0\le x\le \pi$ and find its average value. Indicate the average value on the graph. Draw a rectangle with base $0\le x\le \pi$ and with area equal to the area under the graph of $f\left(x\right)$
16. Graph $f\left(x\right)=\alpha x+\beta ,a\le x\le b,$ where $\alpha ,\beta ,a,b$ are constants and find its average value. Draw a rectangle with base $a\le x\le b$ and with area equal to the area under the graph of $f\left(x\right).$
17. A mailorder company receives 600 cases of athletic socks every 60 days. The numbver of cases on hand $t$ days after the shipment arrives is $I\left(t\right)=600-20{\left(15t\right)}^{\frac{1}{2}}.$ Find the average daily inventory. If the holding for one case is 1/2 cent per day, find the total daily holding cost.
18. Find the average value of $y$ with respect to $x$ for that part of the curve $y={\left(ax\right)}^{\frac{1}{2}}$ between $x=a$ and $x=3a$
19. Find the average value of $y^2$ with respect to $x$ for the curve $ay=b{\left(a^2-x^2\right)}^{\frac{1}{2}}$ between $x=0$ and $x=a.$ Also find the average value of $y$ with respect to $x^2$ for $0\le x\le a$
20. A point moves in a straight line during the time from $t=0$ to $t=3$ according to the law $s=120t-16t^2$
1. Find the average value of the velocity with respect to time for these three seconds
2. Find the average value of the velocity, with respect to the distance $sfor\; these\; three\; seconds$
21. The temperature in a certain city $t$ hours after 9am was approximated by the function $T\left(t\right)=50+14\sin \left(\pi ,t,/,12\right).$ Find the average temperature during the period from 9am to 9pm.

References [PLACEHOLDER]

[BG] A. Braverman and D. Gaitsgory, Crystals via the affine Grassmanian, Duke Math. J. 107 no. 3, (2001), 561-575; arXiv:math/9909077v2, MR1828302 (2002e:20083)

page history