Calculus-Sec-14-7-Solution
14.7 The Substitution Method by SGLee - HSKim, YJLim- SWSun
1. Evaluate 
, where 
is the region enclosed by the lines 
,
and 
.
http://matrix.skku.ac.kr/cal-lab/cal-14-7-1(new).html
as 
.
2. Evaluate 
,
where 
is the region enclosed by the rhombus with vertices 
, 
, 
and 
.
http://matrix.skku.ac.kr/cal-lab/cal-14-7-2(new).html
The region 
is enclosed by the line 
, 
, 
, 
.
Using Changes of Variables, 
, 
as 
, 
Thus,
.
3. Evaluate 
,
where 
is the region enclosed by the triangle with vertices 
, 
and 
.
http://matrix.skku.ac.kr/cal-lab/cal-14-7-3(new).html
The region 
is enclosed by the line 
, 
and 
, 
.
Using Changes of Variables, 
, 
as 
, 
Thus,
.
4. Evaluate the integral 
, where 
is the trapezoidal region with vertices 
, 
, 
and 
.
http://matrix.skku.ac.kr/cal-lab/cal-14-7-4(new).html
, 
, 
.
5. Evaluate the integral 
,
where 
is the trapezoidal region with vertices 
, 
, 
and 
.
The region 
is enclosed by the line 
, 
, 
and 
.
Using Changes of Variables, 
, 
as 
, 
Thus,
.
6. Derive the formula for triple integration in cylinder coordinates.
7. Evaluate 
, where 
is the solid enclosed by the ellipsoid 
,
using the transformation 
, 
and 
.
8. Evaluate 
,
where 
is the region bounded by the graph of 
.
9. Evaluate 
,
where 
is the rhombus region with vertices 
, 
, 
and 
.
10. Evaluate 
,
where 
is the triangle region with vertices 
, 
and 
.
