Download Practice final 1

Math 152
Math 152 - Final Exam Practice #1
1. Find the volume of the solid formed by rotating the region below about the line y = 1.
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2. The Sudbury Neutrino Observatory is a large spherical tank with a 6 meter radius filled with
heavy water. How much work does is take to pump all the water out of the top of the tank?
(Heavy water has a density of 1107 kilograms per cubic meter.)
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3. The curve pictured below is the curve
x = 3t − t3 ,
y = 3t2
√
√
for t-values from − 3 to 3. That is the total arclength of the curve?
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4. Recall that in a circuit containing a battery with voltage E, and resistor with resistance R
and a capacitor with capacitance C, the charge Q on the capacitor is a function of time t
that satisfies the equation
dQ
R
+ CQ = E.
dt
As the circuit operates, the battery warms up. This causes the voltage to vary with time. If
R = 2 ohms,
C = 3 f arads,
E(t) = 12 t + 4 volts,
find the charge Q on the capacitor at time t if the capacitor starts with no charge, i.e.
Q(0) = 0.
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5. Solve the differential equation
dy
xy
=
.
dx
(x − 2)(x + 4)
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6. Consider the sprial r = ln θ.
(a) Write an integral to express the arclength of the curve from θ = 1 to θ = 3. (Do NOT
try to evaluate the integral!)
(b) Approximate the arclength from part (a) using the trapezoid method and ∆θ = 21 .
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7. Calculate
Z
3
4.5 p
6x − x2 dx.