Mathematics 53
I. Evaluate the following integrals. Z sin 2x dx 1. (1 + sin2 x)3 Z √ 2. cos5 x sin xdx Z 3. 2
1
18x3 + 7x2 − 28x + 3 dx 2x + 3
Exercise Set 4
Z 4.
π 2
p cos x − cos3 xdx
− π2
5.
d dx
Z
−x2
x2
1 dt 1−t
II. Do as indicated. x2 + 1 and 5x + y = 1 2 (a) Solve for the points of intersection of the curves enclosing R1 .
1. Given the region R1 bounded by y = 7 − x2 , y =
(b) Set-up the integral for the area of R1 . (c) Set-up the integral for the perimeter of R1 .
2. Given the region R2 bounded by x = −(y + 2)2 + 1 and the line y = x − 1 (a) Solve for the points of intersection of the curves enclosing R2 . (b) Set-up the integral for the volume of the solid generated when R2 is rotated about the line y = 1 (Using Cylindrical Shell Method). (c) Set-up the integral for the volume of the solid generated when R2 is rotated about the line x = −4 (Using Washer Method).