Stepwise problems - Thu. 11:59pm
Partial fractions
01
01
Distinct linear factors
Compute the integral:
Solution
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:
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02
02
Long division first
Compute the integral:
Solution
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):
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03
03
Repeated factor
Compute the integral:
Solution
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:
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Simpson’s Rule
04
01
Simpson’s Rule
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.
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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:
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Regular problems - Sun. 11:59pm
Partial fractions
05
04
Partial fractions - irreducible quadratic
Compute the integral:
Solution
05
(1) Perform long division:
(2) Use
to integrate: Recall formula:
Choose
. Then: The final answer is therefore:
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06
05
Partial fractions - long division
Compute the integral:
Solution
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:
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07
06
Partial fractions - big generic
Give the generic partial fraction decomposition (no need to solve for the constants):
Solution
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:
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- 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
07
Partial fractions - linear and quadratic
Compute the integral:
Solution
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:
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09
08
Partial fractions - repeated factor
Compute the integral:
Solution
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:
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Simpson’s Rule
10
02
Simpson’s Rule for volume by shells
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?
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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:
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11
03
Area of a garden bed
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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: Link to original
