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HW 7.3 MATH 202 (5586023)

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1. Question Details SEssC alc2 7.3.002. [2694100] -

Let S be the solid obtained by rotating the region shown in the figure about the y-axis. (Assume a = 12 and b = 4.)

Sketch a typical approximating shell.

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Find its circumference c and height h.
c(x) =
h(x) =

Use shells to find the volume V of S.
V=

Solution or Explanation
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2. Question Details SEssC alc2 7.3.001. [2693223] -

Let S be the solid obtained by rotating the region shown in the figure about the y-axis. (Assume a = 8 and b = 2.)

Sketch a typical approximating shell.

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What are its circumference c and height h?
c(x) =
h(x) =

Use shells to find the volume V of S.
V=

Solution or Explanation
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3. Question Details SEssC alc2 7.3.005.MI. [2560717] -

Use the method of cylindrical shells to find the volume V generated by rotating the region bounded by the given curves
about the y-axis.

y = 7e−x2 , y = 0, x = 0, x = 1

V=

Sketch the region and a typical shell.

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Solution or Explanation 4/12
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4. Question Details SEssC alc2 7.3.008. [2164437] -

Let V be the volume of the solid obtained by rotating about the y-axis the region bounded y = 25x and y = x2 . Find V
by slicing. 25

V=

Draw a diagram to explain your method.

Find V by cylindrical shells.
V=

Draw a diagram to explain your method.

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Solution or Explanation
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5. Question Details SEssC alc2 7.3.009. [2164899] -

Use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the given
curves about the x-axis.

xy = 3, x = 0, y = 3, y = 5

Solution or Explanation

xy = 3 x = 3 , so
y

5 3 dy
y
V = 2π y

3

= 2π 5 = 2π 3y 5

3 dy

33

= 2π(15 − 9) = 12π

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6. Question Details SEssC alc2 7.3.011. [2164401] -

Use the method of cylindrical shells to find the volume V of the solid obtained by rotating the region bounded by the given
curves about the x-axis.

y = x3, y = 27, x = 0

V=

Sketch the region and a typical shell.

Solution or Explanation
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7. Question Details SEssC alc2 7.3.012. [2163884] -

Use the method of cylindrical shells to find the volume V of the solid obtained by rotating the region bounded by the given
curves about the x-axis.

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x = 12y2 − 4y3, x = 0

V=

Sketch the region and a typical shell.

Solution or Explanation 8/12
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8.

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Question Details SEssC alc2 7.3.013. [2163722] -

Use the method of cylindrical shells to find the volume V of the solid obtained by rotating the region bounded by the given
curves about the x-axis.

x = 1 + (y − 5)2, x = 10

V=

Sketch the region and a typical shell.

Solution or Explanation 9/12
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9. Question Details SEssC alc2 7.3.015.MI. [2560878] -

Use the method of cylindrical shells to find the volume V generated by rotating the region bounded by the given curves
about x = 3.

y = 5x4, y = 0, x = 2

V=

Solution or Explanation
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10. Question Details SEssC alc2 7.3.014. [2164092] -

Use the method of cylindrical shells to find the volume V of the solid obtained by rotating the region bounded by the given
curves about the x-axis.

x + y = 3, x = 4 − (y − 1)2

V=

Sketch the region and a typical shell.

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Solution or Explanation
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11. Question Details SEssC alc2 7.3.016. [2163936] -

Use the method of cylindrical shells to find the volume V generated by rotating the region bounded by the given curves
about the specified axis.

y = 2 x, y = 0, x = 1; about x = −2

V=

Solution or Explanation
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12. Question Details SEssC alc2 7.3.017. [2163817] -

Use the method of cylindrical shells to find the volume V generated by rotating the region bounded by the given curves
about the specified axis.

y = 12x − x2, y = 32; about x = 4

V=

Solution or Explanation
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13. Question Details SEssC alc2 7.3.018. [2164558] -

Use the method of cylindrical shells to find the volume V generated by rotating the region bounded by the given curves
about the specified axis.

y = 8 − x2, y = x2; about x = 2

V=

Solution or Explanation
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14. Question Details SEssC alc2 7.3.019. [2163952] -

Use the method of cylindrical shells to find the volume V generated by rotating the region bounded by the given curves
about y = 8.

27y = x3, y = 0, x = 6

V=

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