Calculus II
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W01 - HW Solutions

01

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(1) Set up integral.

(2) Perform -substitution with and :

(3) Integrate with power rule:

02

(1) Select and considering LIATE:

(2) Apply IBP formula :

Note A: We can change notation to because the value of is arbitrary.

03

(1) Select and considering LIATE:

(2) Apply IBP formula :

(3) Select another and and do IBP again:

(4) Put all together in (A):

Note B: We can change notation to because the value of is arbitrary.

04

You can set up each integral using disks/washers or using shells.

(i) Using washers, obtain:

Using shells, obtain:

(ii) Using disks, obtain:

Using shells, obtain:

05

(a)

(1) Write down shells formula:

(2) Define the cross section region:

Bounded above by . Bounded below by .

Bounded left by . Bounded right by intersection at line .

(3) Define and and :

(4) Plug into shells formula and compute:

(b)

(1) Write down washers formula using :

(2) Rewrite bounding equations in terms of :

(3) Determine region boundary data:

Bounded above by . Bounded below by . Intersection at .

(4) Determine in two components, with the dividing line:

Note that for both regions. These are disks.

(5) Compute the integral:

(6) Why are shells preferable?

  1. Only need one integral.
  2. Don’t need to rewrite boundary equations in terms of .

06

(1) Select and considering LIATE:

(2) Apply IBP formula :

07

(1) Select and considering LIATE:

(2) Apply IBP formula and compute integral:

08

(1) Select and considering LIATE:

(2) Apply IBP formula and compute integral:

(3) Perform -sub with and :

(4) Insert result in Exp. (A):

Note B: We can change to because the inner expression is never negative.

09

(1) Select and considering LIATE:

(2) Apply IBP formula and compute integral:

Therefore:

(3) Repeat. Select and considering LIATE:

(4) Apply IBP formula and compute integral:

(5) Now insert in Eqn. A:

Introduce notational label:

Now use this label in Eqn. A and solve for :

Additional problems

Extra practice

Compute the integral:

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

Solution

(1) Make substitution .

(2) Select appropriate and , and compute and . Use LIATE acronym.

(3) Use the integration by parts formula to evaluate the integral.

(4) 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)

(1) Formula for volume by cylindrical shells

(2) Define cross section region

Bounded above by .

Bounded below by .

Bounded left by line .

Bounded right by line .

(3) Define , the radius of the cylindrical shells.

(4) Define , the height of the cylindrical shells.

(5) Define .

is the limit of here so .

(6) Plug in values to set up integral

(b)

(1) Factor out constants

(2) Expand integrand

(3) Evaluate integral

Extra practice

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

Solution

(1) Choose appropriate and to compute and . Use LIATE acronym.

(2) 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)

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

(2) Define cross section region.

Bounded above by

Bounded below by

Bounded right by intersection at .

Bounded left by .

(3) Define , the outermost radius.

(4) Define , the innermost radius.

(5) Set up integral using derived values.

(b)

(1) Move constant outside and expand integrand.

(2) Evaluate integral.

Extra practice

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

Solution

(1) Choose appropriate and to compute and . Use LIATE acronym.

(2) Use integration by parts formula to set up integral.

(3) Use -substitution.

Set ,

(4) Substitute back for .