Calculus II Final Exam

This is a 3 hour exam. Graphing Calculators not allowed.

Show all your work. Write neatly and large enough to see !!

A/ Integrate the following

(30)

______________________________

B/ Make a diagram for questions 1 - 4.

1) Find the area of the region bounded by y = 6 - x² and y = 3 - 2x .

(5)

2) Use DISK method to find the volume of the solid
obtained by revolving the area bounded by:
 y = x2 + 2, 2y - x - 2 = 0, x = 0, and x = 1 about the x - axis.

(5)

3)Use SHELL method to find the volume of the solid generated by revolving
about the Y-axis the region bounded by
y = 2x - x² and y = 0 .

(5)

4)The base of a solid is the region in the XY-plane bounded by the graph of x² + y² = 16.
Find its volume if every cross section by a plane perpendicular to the X-axis is
an isosceles triangle with base on the XY-plane, with height = the base.

(5)

5)Find the length of the arc along ( y + 1)² = ( x - 4 )³
from point
A (5, 0) to point B (8, 7).

(5)

6)Find z such that z satisfies the conditions of the Mean Value Theorem
for definite integrals if
f(x) = 3x² on the interval from 0 to 3.

(4)

_________________________________________

C/ Find the limits (Show all your work!)

 .

(15)

___________________________________

D/ Evaluate if possible

(12)

_______________________________________

E/ Test for Convergence

1) Test these infinite series. If it converges, find the sum. State the test used.

(8)

2) a) State the 3 requisites for use of the Integral test.

(3)

b) Use the Integral test to prove that the harmonic series, S(1/n) diverges.

(3)

TOTAL (100)

.

.