Chapter 10 Problems

Thirty chapter problems: 10 from the original book and 20 added for more practice.

Read the chapter problems → · Chapter solutions →

1. Original problem

   Derive the formula for the volume of an ellipsoid with a circular cross-section in general terms using the method of discs. Use the image below as a reference.

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   Source ellipsoid reference profile

   An orange upper half-ellipse is drawn above the x-axis and is symmetric about the y-axis. In the illustrated grid it meets the x-axis at x=−3 and x=3 and reaches y=2 on the y-axis.

   A horizontal double-headed arrow from the left endpoint to the origin is labeled a, semi-major axis. A vertical double-headed arrow from the x-axis to the top of the curve is labeled b, semi-minor axis. The source prints the following profile equation.

   $$y=\frac{b\sqrt{a^2-x^2}}{a}$$

   The 3 and 2 values belong to the pictured example. The existing task asks for a general derivation in the labeled axis lengths a and b; the overlay does not substitute those example numbers for the general parameters or give the volume formula.

   

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2. Original problem

   Calculate the volume formed when the area bounded by the curves y=x and y=x³ is rotated about the y-axis using the method of cylinders. The intervals of integration are from x=0 to x=0.75.

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3. Original problem

   Given that the surface area of a sphere is A=4πr², prove the volume of a sphere using spherical shell integration.

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4. Original problem

   A vase that is used for carrying water is 0.75m tall. It is formed by revolving the curve y=x-x⁴ about the x-axis from the interval of x=0.25 to x=1. Using either method, how many cubic meters of water can this vase hold?

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5. Original problem · clarified

   Solve for the volume of the curve $y=x^3$ rotated around the y-axis from x = 1 to x = 2

   Clarification: Use the region bounded by the y-axis, the curve x=y^(1/3), and the horizontal lines y=1 and y=8, then rotate it about the y-axis.

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6. Original problem · clarified

   Your grandma needs a filling in her tooth. Her dentist estimates the shape of her tooth using the curve $y=x^4-3x^2-3$. The range of the curve is from x=-2mm to x=2mm. If her tooth is 5 mm wide, what is the volume of the filling used?

   Clarification: The filling is the cavity below the baseline y=0, extruded through a constant width of 5 mm. On −2≤x≤2, use depth max(0,−(x⁴−3x²−3)); exclude the end pieces above y=0.

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7. Original problem · clarified

   Using the method of washers, find the volume when the region between $x=y^2$ and $x=e^{y-1}$, for 0≤y≤1, is rotated about the y-axis.

   Clarification: The limits 0 and 1 are y-bounds. Use the y-axis rotation stated in the prose; the source widget’s x-axis label conflicts with it. This strip is not the full enclosed leaf.

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   The workspace uses the clarified y-axis problem above.

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8. Original problem

   Derive the formula for the volume of a cone: ​
   a) Using cylindrical shells​
   b) using the washer method

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9. Original problem

   Using either method, calculate the volume of water inside a container that is the shape of the curve y=sin(2x)+1.5 revolved around the x-axis from the interval x=0 to x= $\frac{3\pi }{4}$

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10. Original problem

    Repeat the above problem, but integrate over the interval x=0 to x=2π.

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    Shared source instruction or preceding problem:

    Using either method, calculate the volume of water inside a container that is the shape of the curve y=sin(2x)+1.5 revolved around the x-axis from the interval x=0 to x= $\frac{3\pi }{4}$

    Problem 10 says to repeat the previous problem with new bounds. Reuse the existing readable Problem 9 statement as context, then keep Problem 10’s own bounds.

    

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11. Added problem

    Use cylindrical shells to find the volume when the region $0\le x\le 2$, $0\le y\le 4-x^2$ is revolved about the y-axis.

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12. Added problem

    Revolve the region $1\le x\le 3$, $0\le y\le 1/x$ about the y-axis. Find its volume with shells.

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13. Added problem

    Use shells for the region $0\le x\le 2$, $0\le y\le x^2$ revolved about the vertical line $x=3$.

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14. Added problem

    In the first quadrant, the region is bounded by $x=0$, $y=x^2$, and $y=4$. Revolve it about the x-axis and find its volume using horizontal shells.

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15. Added problem

    Use shells to find the volume obtained by revolving the first-quadrant region bounded by $x=0$, $y=x^2$, and $y=1$ about $y=3$.

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16. Added problem

    A student uses $2\pi {{\int }_0}^1{x}^2(1-x)dx$ for the region $0\le x\le 1$, $0\le y\le 1-x$ revolved about the y-axis. Explain the setup error and find the correct volume.

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17. Added problem

    Find the volume when the region $0\le x\le 2$, $1\le y\le x+1$ is revolved about the x-axis. Use washers.

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18. Added problem

    Revolve $0\le y\le 2$, $0\le x\le y^2$ about the y-axis. Use horizontal discs to find the volume.

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19. Added problem

    Find the volume generated by revolving $0\le x\le 4$, $0\le y\le \sqrt{x}$ about $y=-1$. Use washers.

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20. Added problem

    Use washers to revolve the region $0\le y\le 2$, $y^2\le x\le 4$ about $x=-2$. Find its volume.

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21. Added problem

    Find the volume obtained by revolving the region between $y=0$ and $y=x$ on $-1\le x\le 1$ about the x-axis.

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22. Added problem

    For $0\le x\le 1$, revolve the region $-1\le y\le 2+x$ about the x-axis. A student subtracts $\pi ·1^2$ from each cross-sectional area. Explain why that is wrong and find the volume.

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23. Added problem

    A sphere of radius $R>0$ is centered at the origin. Use discs to find the volume of the portion between the planes $x=0$ and $x=R/2$.

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24. Added problem

    A solid frustum has height $h>0$ and end radii $0\le r\le R$. Its radius varies linearly along its axis. Derive its volume with discs.

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25. Added problem

    A solid has base region between $y=x^2$ and $y=2x$ for $0\le x\le 2$, with coordinates in centimeters. Cross sections perpendicular to the x-axis are squares whose sides equal the vertical gap in the base. Find its volume.

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26. Added problem

    A solid has base disk $x^2+y^2\le 4$. Cross sections perpendicular to the x-axis are semicircles whose diameters are the vertical chords of the base disk. Find the volume. You may use semicircle area $\pi d^2/8$ for diameter d.

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27. Added problem

    A solid has base region $0\le x\le 9$, $0\le y\le \sqrt{x}$. Cross sections perpendicular to the x-axis are equilateral triangles with side length equal to the base’s vertical gap. Given triangle area $A=\sqrt{3}{s}^2/4$, find the volume.

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28. Added problem

    A solid’s cross-sectional area is $A\left(x\right)=x^2$ m² for $0\le x\le 1$ and $A\left(x\right)=4-x$ m² for $1<x\le 3$, where x is measured in meters. Find its volume.

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29. Added problem

    A solid has measured cross-sectional areas of 4, 7, and 5 cm² at axial positions 0, 1, and 3 cm. Estimate its volume on this interval by a trapezoid between each pair of samples.

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30. Added problem

    Revolve the region bounded by $y=x$ and $y=x^2$ on $0\le x\le 1$ about the x-axis. Compute its volume once with washers and once with horizontal shells, showing both sets of radii and bounds.

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