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 .

as

.

2. Evaluate ,

where  is the region enclosed  by the rhombus with vertices  and .

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 .

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 .

.

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 .