MATHEMATICS H161--AU03 MIDTERM ONE
Do all problems. Show your work. (Even if you get an answer just from a calculator, you need to explain why it is correct.)
1. Calculate the following definite integrals.
(a) (5 points) The integral from 0 to pi/2 of the integrand cos3 x.
(b) (5 points) The integral from 0 to 1 of the integrand (x3 multiplied by the square root of 1 Ð x2).
2. Find the following indefinite integrals.
(a) (5 points) The integrand is (2x Ð 1)/square root of (4x2 Ð 4x + 3)
(b) (5 points) The integrand is x/square root of (x - 2)
3. (10 points) Find the area of the region bounded by the straight line y = x + 2 and the parabola y = 4 Ð x2 from x = Ð2 to x = 1.
4. (a) (10 points) The region bounded by the parabola y = 2x2 and the straight line y = 2 is rotated about the y-axis. Find the volume of the resulting solid.
(b) (10 points) The same region as in (a) is rotated about the x-axis. Find the volume of the resulting solid.
5. (10 points) Let f(x) = Ð2x + 1. Prove using an epsilon/delta argument that limit as x approaches 1 of f(x) is Ð1.