MA104 – Midterm Test February 10, 2012 Page 1 of 2
1. Evaluate each of the following integrals. Show all of your work.
[4 marks] (a) cos (x)sin (x)dx
[2 marks] (b) π/4 tan (t)dt
[3 marks] (c) x−4 dx
[3 marks] (d) |1 − sec(x)|dx
[2 marks] (e) 0 sinh(x)dx
[5 marks] 2. Give the Riemann sum argument, using regular partitions, to derive the formula for volumes
of revolution. Assume the area under y = f(x) (where f is a continuous function), above the
x-axis, between x = a and x = b (b > a) is rotated about the x-axis. Make sure all terms are
[5 marks] 3. Determine the exact value of the length of the arc traversed (coverey =by 1 − x 2 from
the point (0,1) to (1/2, 3/4).
[6 marks] 4. Determine the lateral surface area generated by rotating the ay = f(x) = ln(1 − x ) 2 ,
from x = 0 to x = 1/2, about the y-axis. (Note: The problem is changed from the original
[5 marks] 5. Suppose that the function y = f(x), 4 ≤ x ≤ 6, is twice continuously diﬀerentiable on
[4,6] and |f (x)| ≤ 3 on [4,6]. Determine the number n of subdivisions necessary using the
trapezoid rule to ensure the error in approximatin4 f(x)dx is less than 0.01.
6. Compute each of the following integrals, using the method of improper integration. The
majority of marks in each will be for proper form.
[4 marks] (a) 0 xe dx
R1 −1/2 2
[4 marks] (b) 0 x (1 − x) dx
[4 marks] 7. Use the information in the given table about the function y = f(x), 1 ≤ x ≤ 2 and Simp-