PROBLEM 1: Compute the area of the enclosed region bounded by the graphs of the equations , , and .
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SOLUTION 1: Compute the area of the enclosed region bounded by the graphs of the equations , and . Begin by finding the points of intersection of the two graphs. From and we get that
truein
truein
truein
truein
truein
Using vertical cross-sections we get that
truein
AREA
truein
truein
truein
truein
truein
truein
trueintrueintruein
PROBLEM 2: Compute the area of the enclosed region bounded by the graphs of the equations and .
trueintruein
SOLUTION 2: Compute the area of the enclosed region bounded by the graphs of the equations and . Begin by finding the points of intersection of the two graphs. From and we get that
truein
truein
truein
truein
or
truein
Using vertical cross-sections we get that
truein
AREA
truein
truein
truein
truein
truein
truein
truein
truein
.
trueintrueintruein
PROBLEM 3: Compute the area of the enclosed region bounded by the graphs of the equations and .
trueintruein
SOLUTION 3: Compute the area of the enclosed region bounded by the graphs of the equations , , and . Begin by finding the points of intersection of the two graphs. From and we get that
truein
truein
truein
truein
Using vertical cross-sections we get that
truein
AREA
truein
truein
truein
truein
truein
truein
truein
truein
trueintrueintruein
PROBLEM 4: Compute the area of the enclosed region bounded by the graphs of the equations and .
trueintruein
SOLUTION 4: Compute the area of the enclosed region bounded by the graphs of the equations and . Begin by finding the points of intersection of the two graphs. From and we get that
truein
truein
truein
truein
truein
or
truein
Using vertical cross-sections we get that
truein
AREA
truein
truein
truein
truein
truein
truein
truein
truein
truein
truein
truein
trueintrueintruein
PROBLEM 5: Compute the area of the enclosed region bounded by the graphs of the equations and .
trueintruein
SOLUTION 2: Compute the area of the enclosed region bounded by the graphs of the equations and . Begin by finding the points of intersection of the two graphs. From and we get that
truein
truein
truein
truein
truein
, , or
truein
Using vertical cross-sections we get that
truein
AREA
truein
truein
truein
truein
truein
truein
truein
truein
truein
truein
truein
truein
trueintrueintruein
PROBLEM 6: Compute the area of the enclosed region bounded by the graphs of the equations and .
trueintruein
SOLUTION 6: Compute the area of the enclosed region bounded by the graphs of the equations , and . Begin by finding the points of intersection of the two graphs. From and we get that
truein
truein
truein
truein
Using vertical cross-sections we get that
truein
AREA
truein
truein
truein
Let .
truein
Use integration by parts. Let and such that and .
truein
truein
truein
truein
truein
truein
truein
truein
truein
truein
trueintrueintruein
PROBLEM 7: Compute the area of the enclosed region bounded by the graphs of the equations and .
trueintruein
SOLUTION 7: Compute the area of the enclosed region bounded by the graphs of the equations , and . Begin by finding the points of intersection of the two graphs. From and we get that
truein
truein
truein
truein
truein
Using vertical cross-sections we get that
truein
AREA
truein
truein
truein
truein
truein
truein
trueintrueintruein
PROBLEM 8: Compute the area of the enclosed region bounded by the graphs of the equations and .
trueintruein
SOLUTION 8: Compute the area of the enclosed region bounded by the graphs of the equations , and . Begin by finding the points of intersection of the two graphs. From and we get that
truein
truein
truein
truein
or
truein
or
truein
Using horizontal cross-sections such that and we get that
truein
AREA
truein
truein
truein
truein
truein
truein
truein
trueintrueintruein
PROBLEM 9: Compute the area of the enclosed region bounded by the graphs of the equations and .
trueintruein
SOLUTION 9: Compute the area of the enclosed region bounded by the graphs of the equations and . Begin by finding the points of intersection of the two graphs. From and we get that
truein
truein
truein
truein
or
truein
or
truein
Using vertical cross-sections we get that
truein
AREA
truein
truein
truein
Let and .
truein
(Recall from PROBLEM 6 that )
truein
truein
truein
truein
truein
Use integration by parts. Let and such that and .
truein
truein
truein
truein
truein
truein
truein
trueintrueintruein
PROBLEM 10: Compute the area of the enclosed region bounded by the graphs of the equations , and .
trueintruein
SOLUTION 10: Compute the area of the enclosed region bounded by the graphs of the equations , , and . Begin by finding the points of intersection of the two graphs. From and we get that
truein
truein
truein
truein
Using vertical cross-sections we get that
truein
AREA
truein
truein
truein
truein
truein
trueintrueintruein
PROBLEM 11: Compute the area of the enclosed region bounded by the graphs of the equations and .
trueintruein
SOLUTION 11: Compute the area of the enclosed region bounded by the graphs of the equations , , and . Begin by finding the points of intersection of the two graphs.
truein
From and we get that
truein
truein
truein
truein
From and we get that
truein
truein
truein
truein
From and we get that
truein
truein
truein
truein
Using vertical cross-sections we get that
truein
AREA
truein
truein
truein
truein
truein
truein
truein
truein
trueintrueintruein
PROBLEM 12: Compute the area of the enclosed region bounded by the graphs of the equations , , and .
trueintruein
SOLUTION 12: Compute the area of the enclosed region bounded by the graphs of the equations , , and . Begin by finding the points of intersection of the two graphs. From and we get that
truein
truein
Using vertical cross-sections we get that
truein
AREA
truein
truein
Use integration by parts. Let and such that and .
truein
truein
truein
truein
Use substitution. Let
truein
truein
so that
truein
truein
truein
truein
truein
Use integration by parts. Let and such that and .
truein
truein
truein
truein
truein
truein
Use integration by parts. Let and such that and
truein
truein
truein
truein
trueintrueintruein
PROBLEM 13: Compute the area of the enclosed region bounded by the graphs of the equations and .
trueintruein
SOLUTION 13: Compute the area of the enclosed region bounded by the graphs of the equations and . Begin by finding the points of intersection of the two graphs. From and we get that
truein
truein
or
truein
Using horizontal cross-sections we get that
truein
AREA
truein
truein
truein
truein
truein
truein
truein
trueintrueintruein
PROBLEM 14: Compute the area of the enclosed region bounded by the graphs of the equations and .
trueintruein
SOLUTION 14: Compute the area of the enclosed region bounded by the graphs of the equations and . Begin by finding the points of intersection of the two graphs. From and we get that
truein
truein
truein
truein
or
truein
Using horizontal cross-sections we get that
truein
AREA
truein
truein
truein
truein
truein
truein
truein
truein
truein
truein
trueintrueintruein
PROBLEM 15: Compute the area of the enclosed region bounded by the graphs of the equations and .
trueintruein
SOLUTION 15: Compute the area of the enclosed region bounded by the graphs of the equations and . Begin by finding the points of intersection of the two graphs. From and we get that
truein
truein
truein
truein
truein
, , or
truein
Using horizontal cross-sections we get that
truein
AREA
truein
truein
truein
truein
truein
truein
truein
truein
truein
truein