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W03 - HW Solutions

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04

The chart above shows a record of ambient temperatures measured each 15 minutes over 3 hours. Compute the approximate average temperature using Simpson’s Rule. You may use a calculator to resolve the arithmetic in your final expression.

Solution

Recall formula for average value.

Interpret and in the context of the problem.

The chart starts at and goes to . So, we can set and .

Use Simpson’s Rule to approximate integral. Use the given table, and note that .

Plug in expression into average value formula.

05

Compute the integral:

Solution

Rewrite as a difference of two fractions.

Split integral into difference of two integrals.

![warning] Evaluate leftmost integral directly and use the formula .

06

Compute the integral:

Solution

Note that the degree of the denominator is not greater than that of the numerator. Proceed by long division and rewrite integrand.

Perform partial fraction decomposition on . Factor the denominator.

Write out generic PFD. Note that there are two linear factors each repeated once in the denominator.

Multiply out both sides by .

To solve for , plug in . To solve for , plug in .

:

:

Therefore

Rewrite integrand with partial fraction decomposition.

Compute integral.

07

Give the generic partial fraction decomposition (no need to solve for the constants):

Solution

Note that the factor is a difference of squares, and thus can be simplified further.

Write generic PFD.

is a quadratic factor repeated once. The corresponding PFD term would be .

is a linear factor repeated thrice. The corresponding PFD terms would then be .

is a linear factor repeated once. The corresponding PFD term would be .

is a linear factor repeated once. The corresponding PFD term would be .

Putting it all together, we get

08

Compute the integral:

Solution

Begin by rewriting integrand as a partial fraction. Write out generic PFD.

Note that is a linear factor repeated once and is a quadratic factor repeated once.

Multiply both sides by

Expand out right hand side and group like terms.

Set up and solve system of equations. (If you can do linear algebra, it’s easier to set up a matrix and solve, but that is beyond the scope of this class).

Note that from the first equation and from the second equation. Plug these expressions into the last equation to solve for

Plug in the first equation to get .

Plug in the second equation to get .

Rewrite integrand in partial fractions.

Expand out second fraction.

Evaluate integral. Note that for the second fraction, . For the third fraction, use the formula .

09

Compute the integral:

Solution

Rewrite integrand in terms of partial fractions. Write out generic PFD form.

Note that is a linear factor repeated once.

is a linear factor repeated thrice.

Multiply both sides by .

Expand and collect like terms.

Solve system of equations:

Since , we can use this value to obtain .

Thus,

Rewrite integrand in terms of partial fractions.

Evaluate integral, use to evaluate the integral of the last three fractions.

10

Use Simpson’s Rule with to compute the volume of the solid obtained by revolving the pictured region about the -axis. Can you do it without using a calculator?

Recall formula for volume by shells rotated around line parallel to -axis.

Interpret , the radius of the shells/cylinders.

Interpret , the height of the shells/cylinders.

Interpret cross section region.

Bound above by .

Bound left by .

Bound right by .

Bound below by -axis.

Put together integral to approximate.

Create table for Simpson’s rule. Your columns should have at least and . Note that the integrand is not just !

Use Simpson’s Rule formula to approximate integral . Note that here is 1 and the values are the rightmost column .

Use obtained value to approximate volume.