HOMEWORK ASSIGNMENT #6
MATH101.209 - INTEGRAL CALCULUS
Due on March 2, 2010
Last Name ——————————-
Student number ———————-
Problem 1 : Evaluate the following integrals using the table of integrals, substitution rule, integration by parts, integrating trigonometric functions or a combination of these methods
(a) R sec
3
(2 θ ) tan
5
(2 θ ) dθ.
(b) R e e x
2 x +2 e x +2 dx .
(c)
R e
− 3 x cos x dx .
(d) R
2
− 2
( x + 2)
√
4 − x 2 dx .

2
(e) R cos
2 ( 1 x
) sin x 2
2 ( 1 x
) dx .
(f) R x x
4
+1 dx .
(g) R
( x x
2
+1)
6 dx .
(h) R x
2 cos
3
( x
3
) dx .
(i)
R tan
4 x dx .

3
(j) R x
4 ln 3 x dx .
(k) R x e
2 x dx .
Problem 2 : Write a definite integral that gives the volume of the solid of revolution formed by revolving the region bounded by the graph of x = e
− y
2 and the y -axis between y = 0 and y = 1 , about the x -axis.
Problem 3 : Find the average value of f ( x ) = x
3 for integrals tells you about this problem.
sin( x
2
) on the interval [0 ,
√
π ] . What does the mean value theorem

4
Problem 4 : Write a definite integral that gives the volume of the solid of revolution formed by revolving the region bounded by y = x
3
+ x + 1 , y = 1 , and x = 1 about the line x = 2 .
Problem 5 (a) Let R be the region bounded by the graphs of y = e x
, y = e
− x
, and x = ln 2 . Compute the area of R .
(b) Set up but do not evaluate the integral that computes the volume of the solid obtained by rotating this region about the y -axis. .
(c) Evaluate the volume of the solid obtained by rotating this region about the line y = − 1 . Set up but do not evaluate the integral.
Problem 6 : Showing all your work and justifying your method, determine the value of d
2 dt 2
Z e
2 t p u 3
1
+ 1 du
!
.

5
Problem 7 : A 20 foot rope that weighs 8 pounds hangs down 20 feet into a 30 foot hole in the ground. How much work do you do in pulling the rope to the top of the hole?
Problem 8 : How much work is done in winching up 15 feet of chain hanging from a 30 foot tower if the chain weighs
10 lbs/ft?
Problem 9 : Consider a cylindrical tank 15 feet high with a radius of 10 feet. The tank is filled to a depth of 10 feet with fluid that weighs 20 lbs/ft3. Find the work done in pumping all of the fluid out of the tank.
Problem 10 : Consider the region R bounded by y = 2 x and y = x
2 . A solid figure P has R as a base region and cross-sections perpendicular to the x -axis are squares. Express the volume of P as a definite integral. DO NOT
EVALUATE.