Exercise

  1. Assume that $a>b.$ Find the surface area of the ``football'' formed by revolving the ellipse
    MATH
    around the $x$-axis. \lbrack ans: MATH]

  2. Find the area of the surface obtained by rotating about the x-axis that part of the curve $y=e^{x}$ that lies above MATHans:MATH]

  3. One arch of the cycloid given parametrically by the formula MATH is revolved around the x-axis. Find the area of the surface produced.\lbrack ans: $64\pi /3$]

  4. Consider the smallest tin can that contains a given sphere.

    1. Compare the volume of the sphere with the volume of the can.

    2. Compare the surface area of the sphere with the area of the curved side of the can.

  5. The region bounded by $y=1/x$ and the $x$-axis situated to the right of $x=1$ is revolved around the $x-$axis.

    1. Show that its volume is finite but its surface area is infinite.

    2. Does this mean that an infinite area can be painted by pouring a finite amount of paint into this solid.

  6. A cylindrical drinking glass of height $h$ and radius $a,$ full of water, is tilted until the water just covers the base. How much water is left?\lbrack ans: MATH] Can you solve this problem by common sense? Don't use calculus at all.

  7. Find the volume of the solid whose base is the disk of radius $5$ and whose cross sections perpendicular to a fixed diameter are equilateral triangles.\lbrack ans: $500\sqrt{3}/3$]

  8. Find the volume of the region common to two right circular cylinders of radius $1$ whose axis intersect at right angles.

  9. Find the volume of the solid of revolution formed by revolving the region bounded by MATH and the x-axis around

    1. the y-axis,\lbrack ans: MATH]

    2. the x-axis.\lbrack ans: MATH]

  10. Let $R$ be the region below $y=1/(1+x^{2})^{2}$ and above $[0,1].$ Find the volume of the solid produced by revolving $R$ about the y-axis.\lbrack ans: $\pi /2.$]

  11. The region $R$ below MATH and above $[\pi ,10\pi ]$ is revolved about the y-axis to produce a solid of revolution. Find the volume of this solid.\lbrack ans: MATH]

  12. Let $R$ be the region below $y=1/(x^{2}+4x+1)$ and above $[0,1].$ Find the volume of the solid produced by revolving $R$ about the line $x=-2.\qquad $[ans: $\pi \ln 6.$]

  13. When a region $R$ in the first quadrant is revolved about the y-axis, a solid of volume 24 is produced. When $R$ is revolved about the line $x=-3,$ a solid of volume $82$ is produced. What is the area of $R?$

  14. Find the centroid of the region bounded by $y=x^{2}$ and $y=4.\qquad $[ans: $(0,\frac{12}{5})$]

  15. Find the centroid of the region bounded by $y=4x-x^{2}$ and $x-$axis$.\qquad $[ans: $(2,\frac{8}{5})$]

  16. Find the centroid of the region bounded by $y=\sqrt{1+x}$ and $x-$axis, between the lines $x=0$ and $x=3.\qquad $[ans: MATH]

  17. Geologists, when considering the origin of mountain ranges, estimate the energy required to lift a mountain up from sea level. Assume that two mountains are composed of the same type of matter, which weights $k$ pounds per cubic foot. Both are right circular cones in which the height is equal to the radius. One mountain is twice as high as the other. The base of each is at sea level. If the work required to lift the matter in the smaller mountain above the sea level is $W,$ what is the corresponding work for the larger mountain?

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