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W01 - HW Solutions

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© Aleesha Khurram <wma9tt@virginia.edu> 9/2024 – Permitted for use in McMillan’s Calc II only, NOT FOR DISTRIBUTION –

02

Compute the integral:

Solution Select and , and compute and . Use LIATE acronym.

Use the integration by parts formula to evaluate the integral.

05

Consider the region in the -plane, First Quadrant, bounded by the -axis on the left, by on the top, and on the bottom. A 3D solid is given by revolving this region around the -axis.

  • (a) Find the volume of the solid using the method of shells.
  • (b) Attempt to find the volume of the solid using the method of washers/disks. Explain what happens. Why is this harder? (There are two issues!)

Solution (a)

Formula for calculating the volume of a figure using shells.

Define the cross section region.

Bounded above by

Bounded below by

Bounded left by

Bounded right by intersection at line .

Interpret , the radius of shell-cylinder.

Interpret . The height of shell-cylinder equals distance from lower to upper bounding lines:

Interpret .

is the limit of here so .

Use the values to compute the integral.

(b)

Formula for computing volume using washers/disks rotated around line parallel to the axis.

Rewrite the bounds of the cross section in terms of .

Define cross section region.

Bounded above by

Bounded below by

Intersection at .

Define , the outermost radius.

Since varies based on , we have to compute two integrals; one from to , and one from to .

From to , . From to , .

Define , the innermost radius. Since there is no ring in the center, for both integrals.

Compute the integral using derived values.

Concluding thoughts: why are shells preferable?

No need to compute two integrals.

No need to rewrite equations in terms of .

06

Compute the integral:

Number your steps!

Solution

  1. Select appropriate and , and compute and . Use LIATE acronym.
  1. Use the integration by parts formula to evaluate the integral.

Additional problems

Extra practice

Compute the integral:

(Hint: do substitution to get . And then, try both options for , choices.) Number your steps!

Solution

Make substitution .

Select appropriate and , and compute and . Use LIATE acronym.

Use the integration by parts formula to evaluate the integral.

Substitute back for .

Extra practice

Problem A solid is obtained by rotating the region in the first quadrant bounded by curves , , and . about the line .

  • (a) Set up an integral to find the volume of the solid.
  • (b) Evaluate the integral to find the volume of the solid.

Solution (a)

Formula for volume by cylindrical shells

Define cross section region

Bounded above by .

Bounded below by .

Bounded left by line .

Bounded right by line .

Define , the radius of the cylindrical shells.

Define , the height of the cylindrical shells.

Define .

is the limit of here so .

Plug in values to set up integral

(b) Factor out constants

Expand integrand

Evaluate integral

Extra practice

Evaluate the integral. (Use for the constant of integration.)

Solution

Choose appropriate and to compute and . Use LIATE acronym.

Use integration by parts formula to evaluate integral.

Extra practice

A solid is obtained by rotating the area in the first quadrant bounded by the curves and about the -axis. (a) Set up an integral to find the volume of the solid. (b) Evaluate the integral to find the volume of the solid.

Solution (a)

Formula for calculating volume with washers/disks rotated around line parallel to -axis.

Define cross section region.

Bounded above by

Bounded below by

Bounded right by intersection at .

Bounded left by .

Define , the outermost radius.

Define , the innermost radius.

Set up integral using derived values.

(b)

Move constant outside and expand integrand.

Evaluate integral.

Extra practice

Evaluate the integral. (Remember the constant of integration.)

Solution

Choose appropriate and to compute and . Use LIATE acronym.

Use integration by parts formula to set up integral.

Use -substitution.

Set ,

Substitute back for .