Math 1B
Due date: Wednesday 13 April at the beginning of class
Name: _________________________
Exam 2: Take Home Portion
Please only write your name on this page. Show all of your work on separate sheets of paper. Neatly do the problems in
order and start each problem at the top of a page. Staple this page to the front. Show all important work, and justify any
work done by your calculator. All answers should be exact values.
Please do your own work. You may discuss general concepts with others, but not specific features of the problem.
1.
Prove formula #90 in the back of the text:
2x 2 − 1 −1
x 1− x 2
∫ x sin x dx = 4 sin x + 4 + C
−1
by using integration by parts and any necessary trigonometric substitutions.
2.
Use the Midpoint Rule, the Trapezoid Rule, and Simpson’s Rule with n = 8 to estimate the
5 x
e
integral ∫ dx . Show the (exact) terms of each sum, and for your answers round your estimates
x
1
to six decimal places.
3.
A cubical metal tank has a parabolic gate, held in place by bolts and designed to withstand a fluid
force of 250 pounds without rupturing. The liquid to be stored in the tank has a weight density of
60 lb/ft3.
a) What is the fluid force on the gate when the liquid is 2 feet deep?
b) What is the maximum height to which the tank can be filled without exceeding its design
limitation? Note that the tank is 4 feet high: if the tank can be filled, explain why the gate
won’t rupture; otherwise state the depth of fluid at which the gate will rupture.