Name: UIN: TA or Section:
Midterm 4 Math 220 Fall 2008
Please give yourself 1 hour to take this test. Solutions and a grading quide will be posted in a
few days.
1. (13 points) Compute the following integrals
(a) Z1
1 + x2+x1/3+ sin x dx
(b) Zx2
√x3+ 1 + tan(x/2) dx
(c) Ze
2
1
x(ln x)3dx
(d) Ze
1
ex
1 + e2xdx
2. (3 points)Find the derivative of the function G(x) = Rx2
1arctan x dx Hint: First let H(x) be
an antiderivative of arctan x, and use Part I of the Fundamental Theorem of Calculus.
3. (10 points) Precisely state the Fundamental Theorem of Calculus, part II. Then prove it
using the Integral Mean Value theorem. For each “=”, add commentary.
4. (4 points) Use Simpson’s approximation with n= 6 to find the value of R3
1x2. You do not
need to work it out to a single number, but write out fully a sum that could be done with a
4-operator calculator.
5. (10 points) Let Rbe the finite region bounded by y=√x, y = 2,and x= 0.
(a) Sketch the region Rand find the area inside R.
(b) Consider the solid of revolution obtained by rotating Rabout the line x= 4. Sketch
the solid and find the area of a slice at height y.
(c) Find the volume of the solid of revolution in part (b).
6. (10 points) The acceleration due to gravity on planet Mathisfun is -10 feet per second squared.
If you throw a rock upward with velocity 5 feet per second, and it falls for 4 seconds before
hitting the ground, how high was the building? (Ignore air resistance.)
(a) What are the initial position and velocity?
(b) Find the position and velocity as functions of time.
(c) How high was the building?
7. (extra credit, 3 points) Either of the following: Find the flight time and horizontal range of
an object launched at angle π/3 from horizontal with inital speed 98 meters per second (you
are on Earth.) Or: Set up the integral for the work done to fill a cylinder of height 10 and
radius 5 with if the water must be pumped in from the bottom. The density of water is 62.4
pounds per cubic foot.