Shells
01
Shells volume - offset graph,
-axis Consider the region in the first quadrant bounded by the lines
, , , and the curve . Revolve this about the -axis. Find the volume of the resulting solid.
Link to originalSolution
01
(1) Set up integral.
(2) Perform
-substitution with and :
(3) Integrate with power rule:
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IBP
02
Integration by parts - A and T
Compute the integral:
Solution
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
Link to originalto because the value of is arbitrary.
03
Integration by parts - A and L
Compute the integral:
Solution
06
(1) Select
and considering LIATE:
(2) Apply IBP formula
: Link to original
05
Integration by parts - A and I
Compute the integral:
Solution
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
Link to originalto because the inner expression is never negative.
Trig power products
01
Somewhat odd power product
Compute the integral:
Solution
01
(1) Notice odd power on
. Swap the even bunch:
(2) Integrate with
-sub setting and thus : Link to original
02
Tangent and secant both even
Compute the integral:
Solution
02
(1) Notice
. Therefore integrate with -sub setting and : Link to original
05
Tangent and secant mixed parity
Compute the integral:
- (a) Using
. - (b) Using
. Link to originalSolution
07
(a) Select
and thus :
(b)
(1) Select
and thus :
(3) Swap even bunch using
:
(4) Perform
-sub with and integrate: Link to original
Trig subs
01
Trig sub
Compute the definite integral:
Solution
04
(1) Substitute
and thus . Adjust the bounds as follows: Rewrite the integral:
(2) Use power-to-frequency conversion:
Note A: Use
Link to original, then and this equals for .
03
Trig sub
Compute the integral:
Solution
09
(1) Notice
pattern, so we should make use of the identity . Select
and thus . Then: Plug in and simplify:
(We must assume that
for the relevant values of here.)
(2) Use power-to-frequency conversion:
(3) Convert back to terms of
: First draw a triangle expressing
:
Therefore:
For
, use the double-angle identity: Therefore:
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05
Trig sub
Compute the integral:
Solution
11
(1) Notice
pattern, so we should make use of the identity . Select
and thus . Then:
(2) Convert to
and integrate:
(3) Convert back to terms of
: Draw a triangle expressing
:
Therefore
and . Then: Link to original
Partial fractions
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
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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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:
Link to original
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
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?
Link to originalSolution
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:
Link to original
03
Area of a garden bed
Link to originalSolution
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
