1. Setup and evaluate an integral representing the arc lenght of the parametric curve x(t) = t2 , y(t) = t3 ,
for 1 ≤ t ≤ 2.
2. Set up an integral representing the volume of the solid made by rotating the region enclosed by y = x3 ,
y = 8, and x = 0 about the y-axis.
3. Set up an integral representing the volume of the solid made by rotating the region enclosed by y =
2x2 − x3 and y = 0 about the y-axis.
4. Consider a spring with rest length 10 cm. If 40 N is required to stretch the spring from 10 cm to 15 cm,
how much work is required to stretch it from 15 cm to 18 cm?
5. The current through a circuit consisting of a battery, resistor and inductor in series is modeled with the
differential equation
dI
+ 12I = 60
4
dt
with I(0) = 0.
(a) Without solving this initial value problem (IVP) what is the equilibrium solution for I?
(b) Solve for I(t).
6. Using Newton’s Law of Cooling we see the temperature of a can of Coke placed in a fridge change
according to T (t) = 44 + 28e−kt .
(a) Write down a general form of Newton’s Law of Cooling.
(b) How does the temperature change as t → ∞?
(c) What is T (0)?
(d) When does the can reach 50 degrees?
7. For what values of x does the series
∞
P
n=1
(x−3)n
n
converge?
8. Find the Taylor polynomial of degree 7 about x = 0 for the function f (x) = sin x.
9. Does the sequence {an } with an =
n sin n
n2 +1
converge?
10. Do the following series converge or diverge? Why or why not?
(a)
∞
P
(b)
n=1
∞
P
(c)
n=1
∞
P
ln
(d)
n=1
∞
P
en−1
n=1
n2 +1
n3 +1
n3
5n
n
3n+1