MATH 152, Fall 2013
Week In Review
Week 2
1. Evaluate
e
Z
√
1
lnx
dx.
x
2. Compute the area enclosed by the parabola y = x2 and the straight line y = 2x.
3. If f (x) =
Z
0
x2
tan(t2 + 1) dt, then nd f 0 (x).
4. Evaluate the integral
5. Find
Z
1
2
R2 √
−2
4 − x2 dx by interpreting it in terms of areas.
x
dx .
x+1
6. Suppose it is given f (5) = 3, f 0 (5) = 2, f (1) = 4, and f 0 (1) = −1. Compute
R5
1
xf 00 (x)dx.
7. If g(x) =
10
Z
p
t2 + 1dt, then nd g 0 (x).
ln(x)
8. Find the integral
9. If h(x) =
Z
5x
cos(x)
R
x5 ex dx.
2
cos(t2 ) dt, then nd h0 (x).
10. Find the area bounded by the curves x + y 2 = 2 and x + y = 0 .
√
11. Let R denote the region enclosed by the x-axis, the line x = 1, and the curve y = x. Use the method of
cylindrical shells to calculate the volume of the solid obtained by rotating the region R about the line x = 1.
12. Find
Z
√
x3
dx .
x2 + 1
13. An animal population is increasin at a rate of 200 + 50t per year ( where t is measured in years). By how much
does the animal population increase between the fourth and tenth years ?
14. Find the interval on which the curve
Z
y=
0
x
1
dt
1 + t + t2
is concave upward .
15. The velocity function ( in meters per second ) is given for a particle moving on a line as follows
v(t) = t2 − 2t − 8,
1≤t≤6
Find (a) the displacemenet and (b) the distrance traveled by the particle during the given time interval.