Homework 2 – due February 5th
You only need to hand in homework problems 1 and 2. The practice problems are optional.
Homework Problems
1. Find the volume obtained by rotating the region between y = sin(x) and y = cos(x) from x =
x = π around the line y = 2.
Hint: cos(x)2 − sin(x)2 = cos(2x)
2. Define
Z
x2
(x − t) sin2 (t) dt.
G(x) =
1
Find G0 (x), the derivative of G(x).
Practice problems
1. Let f (x) be a continuous function on [0, 5] such that
f (x) + f (x − 1) =
Find
R2
0
q
√
x + 1.
f (x) dx.
2. Let f (x) be a continuous function.
Define
R1
f (x − t) dt and
0
R x2
(b) G(x) = x f (xt) dt, x > 0.
(a) G(x) =
In each case find the derivative G0 (x).
3. Let f (x) be a continuous functions which satisfies the following equation
e
R x−1
0
f (t) dt
Z
+ sin
0
for every real number x.
Find f (0).
1
x−1
2
f (t) dt = ex
π
2
to