Math 166Z Homework # 4.5
Ungraded
1. Evaluate the following Integrals:
Z 4 √
x
dx
(a)
1 x+1
Z √
3
x+1
√
(b)
dx
3
x−1
Z
dx
(c)
(Hint: try completing the square)
(x2 + 2x + 2)2
Z
dy
(d)
4
y − y2
Z 2 3
x + x2 − 12x + 1
(e)
dx
x2 + x − 12
0
Z
1
(f)
dx
(x − 1)3
Z
1
dx
(g)
x(x + 1)(2x + 3)
Z 1
r
dr
(h)
2
0 r + 4r + 4
2. Choose 5 problems at random from the last page of the document
StrategiesForIntegration Stewart.pdf posted on WebCT. Try to do them without looking
at your notes or book.
3. A tank has the shape of an inverted circular cone with a height of 20m and a base radius
of 8m. It is filled with water to a height of 16m. Find the work required to empty
the tank by pumping all of the water to the top of the tank. (The density of water is
1000kg/m3 )
4. Find the centroid of the region bounded by the curves y = sin x, y = cos x, x = 0, x = π4 .
Z
5. Evaluate the integral tan3 t sec4 t dt
(a) Once you have done the integral, can you think of another way to do it? (i.e. can
you get it to work choosing a different u?)
(b) Your answers will look different using the two different methods. Convince yourself
they are the same. (What are some different ways you could do this?)