Exercise 14: Iterated Integral for
Region
: Bounded by
,
, and
(a) Vertical Cross-Sections (Integrate
.
first, then
1. Determine the bounds:
The curves intersect where
For
, ranges from
2. Set up the iterated integral:
(b) Horizontal Cross-Sections (Integrate
)
, so
to .
first, then
1. Determine the bounds:
For
, ranges from
2. Set up the iterated integral:
.
)
to
.
Answer:
(a)
(b)
Exercise 20: Sketch and Evaluate the Integral
Integral:
1. Sketch the Region of Integration:
Bounds:
and
The region is between
and
.
from
to
2. Evaluate the Integral:
Using the identity
:
Answer:
Exercise 50: Reverse Order of Integration and Evaluate
Integral:
1/ 2
.
1. Sketch the Region of Integration:
Bounds:
and
The region is bounded by
,
.
, and
2. Reverse the Order of Integration:
For
, ranges from
The reversed integral is:
to
.
3. Evaluate the Integral:
Answer:
Summary of Answers
1. Exercise 14:
(a)
(b)
2. Exercise 20:
3. Exercise 50:
2/ 2
to
.