Calculus II
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W03 - HW Solutions

01

(1) Write the partial fractions general form equation:

(2) Solve for constants.

Cross multiply:

Plug in , obtain so .

Plug in , obtain so .

(3) Integrate each term:

02

(1) Numerator degree is not smaller! Long division first:

Now this already has the form of a partial fraction decomposition, so we proceed directly to integration.

(2) Integrate using power rule (with log):

03

(1) Write the partial fractions general form equation:

(2) Solve for constants.

Cross multiply:

Plug in , obtain .

Plug in , obtain .

Plug in , obtain:

(3) Integrate each term:

Optional simplification:

04

(1) Recall the formula for the average value of over :

Here and :

(3) Use in Simpson’s Rule:

(4) Plug into average value formula:

05

(1) Perform long division:

(2) Use to integrate:

Recall formula:

Choose . Then:

The final answer is therefore:

06

(1) Numerator degree is not smaller! Long division first:

(2) Factor denominator:

(3) Write the partial fractions general form equation (for the second term):

(4) Solve for constants:

Cross multiply:

Plug in , obtain so .

Plug in , obtain so .

(5) Integrate by terms:

07

(1) Observe that :

On the other hand, cannot be factored further. (Its zeros are imaginary.)

Now all denominator factors are either linear or irreducible quadratic.

(2) Write the partial fractions general form equation:

(3) Notice a few things:

  • Quadratic acquires linear term on top
  • Linear is to 3rd power so it has repetition up to 3rd power
  • Linear and are only to 1st power.

08

(1) Denominator has degree 3, numerator has degree 2, therefore long division is not necessary.

(2) Write the partial fractions general form equation:

Notice that is an irreducible quadratic (cannot be factored). So we have:

(3) Solve for constants:

Cross multiply:

Plug in , obtain:

Expand RHS:

Comparing terms, obtain: and thus .

Comparing constant terms, .

(4) Integrate by terms:

Note A: For the last term, use the formula:

09

(1) Write the partial fractions general form equation:

Observe that appears in degree 3 in the integrand, so we have one term for each power up to 3 in the partial fraction decomposition.

(2) Solve for constants:

Cross multiply:

Plug in , obtain so .

Plug in , obtain .

Plug in , obtain:

Plug in , obtain:

(3) Integrate each term:

Optional simplification:

10

(1) Recall shells formula:

(2) Interpret:

Bounded above by . Bounded below by -axis.

Bounded left by . Bounded right by .

Obtain:

(3) Create table of values to apply Simpson’s Rule:

(4) Recall Simpson’s Rule formula:

Here since in this formula represents the integrand values.

Note that .

Plug in:

Therefore:

Therefore:

11

(1) Set up integration:

Set at the left upper corner, with extending to the right, extending downwards. Then:

(2) Create table of values:

(3) Recall Simpson’s Rule formula:

Here and .

Thus:

(4) Compute cubic yards from known surface area:

Mulch is deep, so the volume is: