Download Calculus II Test III: Integration, Improper Integrals, Area, Volume and Length and more Exams Calculus in PDF only on Docsity!
MA 126-7B, CALCULUS II
October 31, 2011
Name (Print last name first):..........................................
Student Signature:...................................................
TEST III
Closed book - No calculators!
PART I
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^2 − 4
dx.
Answer:.....................
Question 2
Determine whether the improper integral is convergent or divergent. Evaluate the integral if it is convergent. (^) ∫ (^) ∞
1
√ (^4) x 2 dx
Answer:.....................
Question 3
Find the area of the region bounded by the line y = −x + 1, the x-axis, the y-axis and the vertical line x = −1.
Answer:..................
Question 4
Find the area of the region bounded by the curve y = x^3 , the vertical line x = 0 and the horizontal line y = 8.
Answer:..................
Question 5
Find the volume of the solid obtained by rotating the curve y = x^2 about the x-axis for 0 ≤ x ≤ 2.
Answer:..................
Question 6
Find the length of the arc of the circular helix with vector equation r(t) = 〈 2 cos(t),
5 t, 2 sin(t) 〉 when 0 ≤ t ≤ 1.
Answer:..................
Problem 2
(a) Determine whether the (improper) integral ∫ (^) ∞
0
xex^ dx
is convergent or divergent. Evaluate the integral if it is convergent.
(b) Find the area of the region enclosed by the parabola y = 4x − x^2 and the line y = 2x. (Hint: Sketching the region might prove useful here!)
Problem 3
(a) Find the area of the region enclosed by the line y = x−1 and the parabola y^2 = 2x+6.
(b) The region enclosed by the curves y =
x and y = x is rotated about the horizontal line y = −1. Find the volume of the solid obtained in this way.
Problem 5
This problem has two separate questions. (Answer each question!)
(a) Find the length of the arc of the circular helix with vector equation r(t) = 〈 4 cos(t), 3 t, 4 sin(t) 〉 when 0 ≤ t ≤ 1.
(b) A rectangular swimming pool 3 m long, 1 m wide, and 1 m deep is full of water. Find the work needed to pump all the water out over the side. (Use the fact that the density of the water is 1,000 kg/m^3 and g ≈ 10 m/s^2 .)
SCRATCH PAPER
(Scratch paper will not be graded!)
