Worksheet for Calc. II - Review for Midterm
Sections 5.1 - 5.6, 7.1 - 7.6
1. Evaluate the following integrals:
R
2
(a) √xx2 +4 dx
(b)
R
sin(3x) sin(5x) dx
(c)
R
dt
(t2 −6t+13)2
(d)
R
z 5 ez dz
(e)
R
4−y
y(y 2 +2)2
(f)
R
x3 +1
x2 −4
(g)
R
e−t sin(2πt) dt
(h)
R
tan3 θ sec4 θ dθ
2
dy
dx
2. Goofus runs up to Gallant one day after their Calculus class.
“Gallant!” cries Goofus, “I just discovered a new proof that will change the face of
mathematics forever!”
Gallant replies, “Really? What is it?”
Goofus then responds, “I have an infallible proof that shows that 0 = 1!”
“Um, okay,” says Gallant skeptically, “Are you sure it’s correct?”
“Yeah, I’m pretty sure,” Goofus confidently replies, “Here, let me walk you through
it...”
R
“So when solving the indefinite integral x1 dx, we can use integration by parts. Set
u = x1 and dv = dx. Then du = − x12 dx and v = x. Putting all of these parts back
into the integration-by-parts formula,
Z
Z
u dv = uv − v du,
R
R
R
we see that x1 dx = x1 x − x − x12 dx = 1 + x1 dx. Subtracting the integral from
both sides of the equation, and we get 0 = 1. End of proof.”
Gallant lets out a big sigh. What is wrong with Goofus’ logic?
1
3. Let R be the region bounded by the curves y = 2x2 − 7 and y = x2 + 3x − 3.
(a) Sketch R.
(b) Find the area of R.
(c) Find the volume of the solid generated by revolving R about the line y = −7.
(d) Find the volume of the solid generated by revolving R about the line x = 4.
4. Let C be the curve parameterized as
1
y(t) = t2 ,
2
x(t) = 4t,
for 0 ≤ t ≤ 3.
(a) Find the arc length of C.
(b) Find the surface area of the solid generated by revolving the region bounded by
C about the x-axis.
5. Bruno the Brawny is a performer for Dr. Peters’ Traveling Mathematical Circus.
Bruno specializes in balancing large, heavy objects on top of his head, and he is looking
forward to balancing a jukebox for his next show. The
√ shape of the base of the jukebox
is the same as the region bounded by the curve y = 4 − 2x2 and the x-axis. Assuming
that the jukebox’s weight is uniformly distributed on the base, at what position on the
base should Bruno rest the jukebox on his head in order to balance it?
6. A tank which is 20 feet long has semicircular ends with radii of 5 feet, and it is filled
with water. How much work is required to pump all the water out from the very top
of the tank? (Remember that the density of water is 62.4 lb/ft3 )
2
Solutions
(c)
√
√
1
2 + 4 − 2 ln | x2 +4+x | + C
x
x
2
2
1
1
sin(2x) − 16 sin(8x) + C
4
1
t−3
arctan 12 (t − 3) + 18 t2 −6t+13
+
16
(d)
1 4 z2
z e
2
1. (a)
(b)
2
2
− z 2 ez + ez + C
1
2
2
(e) ln |y| − ln(y + 2) +
(f)
7
4
1
6
1− 41 y
y 2 +2
−
1
4
√
2 arctan
√y
2
+C
ln |x + 2| + 94 ln |x − 2| + 12 x2 + C
(g) − 4π21+1 e−t sin(2πt) −
(h)
C
2π
e−t
4π 2 +1
cos(2πt) + C
sec6 θ − 14 sec4 θ + C
2. Hint: Think about how the integration-by-parts formula is derived.
3. (a)
(b)
(c)
(d)
4. (a)
(b)
125
≈ 20.833
6
2375
π ≈ 1, 243.547
6
625
π ≈ 163.625
12
15
+
2
122
π
3
8 ln(2) ≈ 13.045
≈ 127.758
8
5. 0, 3π
≈ (0, 0.849)
6. 104, 000 foot-pounds
3