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MA 126-6D, CALCULUS II
March 31, 2008
Name (Print last name first):..........................................
Student Signature:...................................................
TEST III
No calculators are permitted!
PART I - Basic Skills
Each question is worth 5 points.
Part I consists of 6 questions. Clearly write your answer (only) in the space provided after each question. You do not need not to show your work for this part of the test. No partial credit is awarded for this part of the test!
Question 1
Evaluate the indefinite integral
x x − 6
dx.
Answer:.....................
Question 2
Evaluate the indefinite integral
3 x^2 sin(x^3 ), dx.
Answer:.....................
Question 3
Determine whether the improper integral is convergent or divergent. Evaluate the in- tegral if it is convergent. (^) ∫ ∞ 1
x
dx
Answer:..................
Question 4
Determine whether the improper integral is convergent or divergent. Evaluate the in- tegral if it is convergent. (^) ∫ ∞ 0
2 x (x^2 + 2)^2 dx
Answer:..................
Question 5
Find the area of the region bounded by the parabola y = x^2 + 1, the horizontal line y = 0, and the vertical lines x = 0 and x = 1.
Answer:..................
Question 6
Find the volume of the solid obtained by rotating the curve y = x^3 about the y-axis for 0 ≤ y ≤ 1.
Answer:..................
Problem 2
(a) Evaluate the indefinite integral (^) ∫ x − 2 x^3 + x dx.
(b) Make a substitution to express the integrand as a rational function and then evaluate the integral ∫ (^16)
9
x x − 4 dx (Set u =
x, and note that x = u^2 .)
Problem 3
(a) Determine whether the (improper) integral ∫ (^) ∞
e
x(ln x)^3
dx
is convergent or divergent. Evaluate the integral if it is convergent.
(b) Find the exact area of the region between the graph of the function f (x) = 4x^3 e−x^4 and the x-axis when 0 ≤ x < ∞.
Problem 5
(a) Find the volume of the solid obtained by rotating about the x-axis the region bounded by the curve y =
x, the horizontal line y = 0 and the vertical lines x = 0 and x = 2.
(b) Find the volume of the solid obtained by rotating about the x-axis the region bounded by the curve y = (^1) x , the horizontal line y = 0 and the vertical lines x = 1 and x = 3.
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