Homework 11: 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: 24 February 2010

Problem A. Problems by washers.

1. Show that the volume of a right circular cylinder of radius $r$ and height $h$ is $\pi r^{2}h$ by using the washer method.
2. Show that the volume of a right circular cone of radius $r$ and height $h$ is $\frac{1}{3}\pi r^{2}h$ by using the washer method.
3. Show that the volume of a sphere of radius $r$ is $\frac{4}{3}\pi r^3$ by using the washer method.
4. Find the volume generated when the area bounded by the lines $x+y=2,x=0,y=0$ is rotated about the $x$-axis.
5. Find the volume generated when the area bounded by $y=\sin x,0\le x\le \pi ,$ and $y=0$ is rotated about the $x$-axis.
6. Find the volume generated when the area bounded by $y=x-x^2$ and $y=0$ is rotated about the $x$-axis.
7. Using integration find the volume generated by rotating the triangle with vertices at (0,0), $\left(h,0\right)$ and $\left(h,r\right)$ about the $x$-axis.
8. Using integration find the volume generated by rotating the triangle with vertices at (0,0), $\left(h,0\right)$ and $\left(h,r\right)$ about the $y$-axis.
9. A hemispherical bowl of radius $a$ contains water to a depth of $h.$
1. Find the volume of water in the bowl.
2. Water runs into a hemispherical bowl of radius 5 ft at the rate of 0.2 ${\mathrm{ft}}^3/\sec$. How fast is the waterlevel in the bowl rising when it is 4ft deep?
10. Find the volume generated when the area bounded by $y=-3x-x^2$ and $y=0$
is rotated about the $x$-axis.
11. Find the volume generated when the area bounded by $y=x^2-2x$ and $y=0$ is rotated about the $x$-axis.
12. Find the volume generated when the area bounded by $y=x^3,x=2$ and $y=0$ is rotated about the $x$-axis.
13. A football has a volume that is approximately the same as the volume generated by rotating the area inside the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}$ (where $a$ and $b$ are constants) about the $x$-axis. Find thew volume generated.
14. The volumes of a certain solid generated by planes perpendicular to the $x$-axis are circles with diameters extending from the cirve $y=x^2$ to the curve $y=8-x^2.$ The solid lies between the points of intersection generated by the two curves. Find its volume.
15. The base of a certain solid is the circle $x^2+y^2=a^2.$ Each plane section of the solid ut out by a plane perepndicular to the $x$-axis is a square with one edge of the square in the base of the solid. Find the volume of the solid.
16. Find the volume generated when the area bounded by $y=x^4,x=1$ and $y=0$ is rotated about the $x$-axis.
17. Find the volume generated when the area bounded by $y=\sqrt{\cos x},0\le x\le \frac{\pi }{2}$ and $x=0,y=0$ is rotated about the $x$-axis.
18. Find the volume generated when the area bounded by $y=\sqrt{x},y=2$ and $x=0$ is rotated about the $y$-axis.
19. Two great circles, lying in planes that are perpendicular to each other are marked on a sphere of radius $a.$ A portion of the sphere is shaved off so that any plane section of the remaining solid, perependicular to the common diameter of the two great circles, is a square with vertices on these circles. Find the volume of the solid that remains.
20. The base of a solid is the circle $x^2+y^2=a^2.$ Each plane section of the solid cut out by a plane perependicular to $y$-axis
is an isoceles right triangle with one leg in the base of the solid. Find the volume.
21. The base of a solid is the region between the $x$-axis and the curve $y=\sin x$ between $x=0$ and $x=\frac{\pi }{2}.$ Each plane section of the solid perpendicular to the $x$-axis is an equilateral triangle with one side in the base of the solid. Find the volume.
22. Find the volume generated when the area bounded by $y=\sqrt{x},y=2$ and $x=0$ is rotated about the line $y=2.$

Problem B. Finding volumes by cylindrical shells.

1. Show that the volume of a right circular cylinder of radius $r$ and height $h$ is $\pi r^{2}h$ by using cylindrical shells.
2. Show that the volume of a right circular cone of radius $r$ and height $h$ is $\frac{1}{3}\pi r^{2}h$ by using cylindrical shells.
3. Show that the volume of a sphere of radius $r$ is $\frac{4}{3}\pi r^3$ by using cylindrical shells.
4. A hole of diameter $a$ is bored through the center of a sphere of radius $a.$ Find the remaining volume.
5. Find the volume of the bagel produced by rotating the circle $x^2+y^2=a^2$ about the line $x=b,\left(b\ge a\right)$
6. Find the volume generated by rotating the area bounded by the curves $x+y=2,x=0$ and $y=0$ about the $x$-axis by using cylindrical shells.
7. Find the volume generated by rotating the area bounded by the curves $x=2y-y^2$ and $x=0$ about the $x$-axis by using cylindrical shells.
8. Find the volume generated by rotating the area bounded by the curves $y=3x-x^2$ and $y=x$ about the $x$-axis by using cylindrical shells.
9. Find the volume generated by rotating the area bounded by the curves $y=x,y=1$ and $x=0$ about the $x$-axis by using cylindrical shells.
10. Find the volume generated by rotating the area bounded by the curves $y=x^2$ and $y=4$ about the $x$-axis by using cylindrical shells.
11. Find the volume generated by rotating the area bounded by the curves $y=3+x^2$ and $y=4$ about the $x$-axis by using cylindrical shells.
12. Find the volume generated by rotating the area bounded by the curves $y=x^2+1$ and $y=x+3$ about the $x$-axis by using cylindrical shells.
13. Find the volume generated by rotating the area bounded by the curves $y=4-x^2$ and $y=2-x$ about the $x$-axis by using cylindrical shells.
14. Find the volume generated by rotating the area bounded by the curves $y=x^4,x=1$ and $y=0$ about the $y$-axis by using cylindrical shells.
15. Find the volume generated by rotating the area bounded by the curves $y=x^3,x=2$ and $y=0$ about the $y$-axis by using cylindrical shells.
16. Find the volume generated by rotating the triangle with vertices (1,1), (1,2) and (2,2) about the $x$-axis by using cylindrical shells.
17. Find the volume generated by rotating the triangle with vertices (1,1), (1,2) and (2,2) about the $y$-axis by using cylindrical shells.
18. Find the volume generated by rotating the area bounded by the curves $x=y-y^3,y=0,x=0$ and $y=1$ about the $x$-axis by using cylindrical shells.
19. Find the volume generated by revolving the area bounded by $y=\sqrt{x},x=0$ and $y=2$ about the $x$-axis.
20. Find the volume generated by revolving the area bounded by $y=\sqrt{x},x=0$ and $y=2$ about the line $x=4.$

Problem C. Practical volumes

1. The cross section of a solid in any plane perpendicular to the $x$-axis is a circle having diameter $\mathrm{AB}$ with $A$ on the curve $y^2=4x$ and $B$ on the curve $x^2=4y.$ Find the volume of the solid lying between the points of intersection of the curves.
2. The base of a solid is the area bounded by $y^2=4ax$ and $x=a.$ Each cross section perendicular to the $x$-axis is an equilateral triangle. Find the volume of the solid.
3. Find the volume of the slice obtained by cutting the sphere of radius $r$ if the the slice has thickness $h$ at its thickest point.
4. Find ther volume left after slicing the top off a right circular cone of the cone has radius $r$ and height $h$ and after slicing the top off what's left has height $b.$
5. Find the volume of a tetrahedron, where each slice of the tetrahedron is an equilateral triangle with side length $a.$
6. The base of a solid is a circle of radius $r$ and the cross sections perpendicular to the base are squares. Find the volume.
7. The base of a solid is the ellipse $9x^2+4y^2=36.$ Cross sections perpendicular to the $x$-axis are isoceles right triangles with hypoteneuse in the base. Find the volume.
8. The base of a solid is the parabolic region $\left\{\left(x,y\right)|x^2\le y\le 1\right\}.$ Cross sections perpendicular to the $y$-axis are equilateral triangles. Find the volume.
9. Find the volume common to two spheres, each with radius $r$ where the center of each sphere is on the surface of the other.
10. Find the volume common to two circular cylinders of radius $r$ such that the axes of the cylinders intersect at right angles.
11. In 1715 Kepler published a book, $\mathrm{Stereometria}\mathrm{doliorum},$ which explained how to find the volumes of barrels. A barrel of height $h$ and maximum radius $R$ is constructed by rotating the parabola $y=R-c{x}^2,-\frac{h}{2}\le x\le \frac{h}{2},$ where $c$ is a positive constant.
1. Show that the radius of each end of the barrel is $r=R-c\frac{x^2}{4}.$
2. Show that the volume of the barrel is $\left(\frac{1}{3}\right)\pi h\left(2R^2+r^2-\left(\frac{1}{40}\right)c^2{h}^4\right).$
12. Suppose that you are given two spherical balls of wood, one of radius $r$ and a second one of radius $R.$ A circular hole is bored through the center of each ball and the resulting napkin rings hve the same height $h.$ Which napkin ring contains more wood? How much more?
13. A right circular cone with height 1 meter and base radius $r$ is to be separated into three pieces of equal volume by cutting twice parallel to the base. A what heights should the cuts be made?
14. A drinking cup is filled with water and has the shape of a right circular cone with height $hand\; semivertical\; angle$ \theta .$A\; ball\; is\; placed\; in\; the\; cup\; displacing\; some\; of\; the\; water.\; What\; is\; the\; radius\; of\; the\; ball\; that\; causes\; the\; greatest\; volume\; of\; water\; to\; spill\; out\; of\; the\; cup?$

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)

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