Prof. Brian Harbourne's old exam
- [1] Use the substitution u = 4 - 2x to convert the
integral, from x = 0 to x = 1 of 2x(4 - 2x)1/2 with
respect to x, into an integral in terms of
u only. You do not need to evaluate the integral.
- [2] Let R be the region bounded by the curves
y = x1/2, y = 2 - x, and y = 0.
Sketch the region R, and express its area in terms
of one or more integrals. You do not need to evaluate the integrals.
- [3] Let R be the region bounded by the curves y = 1, x = 1,
and y = 100/x2. Sketch the region R, and write
down an integral for the volume of the solid formed by
revolving R about the y-axis. You donot need to evaluate the integral.
- [4] A force of 10 pounds stretches a spring 2 inches. Set upan
integral for the work done in stretching the spring 3 inches
beyond its natural length. You do not need to evaluate the
integral, but indicate what units of work the integral will
evaluate to.
- [5] Baby bear's porridge is 150F when Mother bear sets it out
at 9:00 am. The den is a comfy (for bears) 50F. At 9:10 am,
the porridge is 120F, still too hot for Baby bear, who likes
porridge to be 99F. What time will it be when Baby bear's
porridge is just right? (Assume the porridge cools according
to Newton's Law of cooling.)
Prof. Bo Deng's old exam
- 1.
- (a) Integrate x(1 + 4x2) -1/2 with respect to x.
- (c) Integrate x(4 + x) -1/2 with respect to x,
from x = 0 to x = 5.
- 2. Let R be the region enclosed by x = 1 + y2, y = x - 1.
- (a) Find (but don't evaluate) an integral whose value gives
the exact area of the region R.
- (b) Find (but don't evaluate) an integral whose value gives
the volume of the solid obtained by revolving the region R
about the y-axis, using the method of shells.
- (c) Find (but don't evaluate) an integral whose value gives
the volume of the solid obtained by revolving the region
R about the vertical line x = 0, using the method of
cylindrical washers.
- 3. A water tank is in the shape of a right circular cone
of altitude 10 feet and base radius
5 feet, with its base at the ground. If the tank is
full, find an integral but do not evaluate for the work
done in pumping the top 5 feet of the water out of the top.
- 5. An underwater observatory has a circular hatch
of 1 foot radius on its (vertical) wall
with the hatch's top 20 feet below the surface. Write down but
do not evaluate an integral whose value is the force that is
required to hold the hatch in place against the water.
Prof. Mohammad Rammaha's old exam
- 1.
- a. Integrate (sin x)/(3 + cos x) with respect to x.
- b. Integrate (ln x)7/x with respect to x
from x = 1 to x = e.
- c. Integrate x -1/2 sec2(x1/2)
with respect to x.
- 2. Let R be the region enclosed by y = x4,
y = x1/2 from x = 0 to x = 1.
- a. Find (but don't evaluate) an integral whose value gives the
exact area of the region R.
- b. Find (but don't evaluate) an integral whose value gives the
volume of the solid obtained by revolving the region R about
the x-axis, by using the method of slicing.
- c. Find (but don't evaluate) an integral whose value gives the
volume of the solid obtained by revolving the region R about
the y-axis, by using the method of cylindrical shells.
- 3. A tank has a square base whose length is 5 feet and
rectangular sides of height 3 feet. Assume that the tank is
filled with water weighing 62.5 lb/ft3.
- a. Find a Riemann sum whose value approximates the work
required to pump all of the water over the top of the tank.
- b. Write down but do not evaluate an integral whose value is
exactly the work required to pump all of the water over the
top of the tank.
- c. Write down but do not evaluate an integral whose value is
exactly the force exerted by the water on one side of the tank.
- 4. Find f-1(x) if y = f(x) = 4 + 3e2x.
- 5. This question deals with the function f(x) = 2x3 + 5x - 1.
- a. Show that f-1 exists.
- b. Find the equation of the tangent line to the function
y = f-1(x) at the point (6, 1).
- 6. Let y(t) be the amount of radioactive element present
at time t >= 0, and assume
that y(t) satisfies the equation: dy/dt = -0.3y.
Write down the exact form of y(t) and find the half-life
of the radioactive element.