practice_prelim1_fa06 - 5 Find the length of the curve...

Practice Prelim 1, Math 191, Fall 2006
No calculators. Show your work. Clearly mark each answer.
This practice prelim may be longer than the actual prelim.
1. The max-min inequality for definite integrals is helpful in the following. [10]
(a) Find upper and lower bounds for
Z
2
0
x
p
1 + x
2
dx .
(b) Rewrite the integral of part (a) by using the substitution u = x
2
. Find upper and lower bounds
for the resulting integral.
2. Write the sum
n X
k =1
n
3
+ k
3
n 4
as a Riemann sum in order to evaluate lim
n →∞
n X
k =1
n
3
+ k
3
n 4
. [10]
3. Evaluate: [20]
(a)
Z
1
- 1
t cos(sin t ) dt (b)
d
dy
Z
4 y
0
sin t cos t dt
(c)
Z
√
3 v - 1 dv (d)
Z
t
√
1 - 4 t 2
dt
4. Consider the region described by y ≤
1
x 2
, y ≤ 8 x , and y ≥
x
2
16
. [15]
(a) Compute its area.
(b) Find its average height.