- [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 = x
^{1/2}, 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/x
^{2}. 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.)

- 1.
- (a) Integrate x(1 + 4x
^{2})^{ -1/2}with respect to x. - (c) Integrate x(4 + x)
^{ -1/2}with respect to x, from x = 0 to x = 5.

- (a) Integrate x(1 + 4x
- 2. Let R be the region enclosed by x = 1 + y
^{2}, 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.

- 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}sec^{2}(x^{1/2}) with respect to x.

- 2. Let R be the region enclosed by y = x
^{4}, y = x^{1/2}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/ft
^{3}.- 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 + 3e^{2x}. - 5. This question deals with the function f(x) = 2x
^{3}+ 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).

- a. Show that f
- 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.