Mathematics 136, section 2 – Advanced Placement Calculus Exam 3 – December 2, 2009

Directions

Do questions I - V. You must show all work done to justify your answers in your solutions. If you use an entry from the extract of the table of integrals, identify it by number. There are 100 regular points and 10 Extra Credit points. Do all work in the blue exam booklet. Good “hunting!”

I. Let R be the region in the plane bounded by y = e^{x}, y = x/2, x = 0, and x = 1.

A) (5) Sketch the region R.

B) (5) Find the area of R.

C) (10) Find the volume of the solid generated by rotating R about the x-axis.

D) (5) Set up, but do not evaluate, the integral or integrals to compute the volume of the solid generated by rotating R about the y-axis.

II. A) (10) Integrate by parts: ∫ xe^{−3x} dx.

B) (10) Does the integral ∫_{0}^{∞} xe^{−3x} dx converge? If so, find its value.

III. (15) A water tank has the shape of an inverted cone (that is, the “point” of the cone is at the bottom) with height 10 meters and radius 3 meters. If the tank is full of water (density 1000kg per cubic meter), find the work done in pumping all the water out the top of the tank. The acceleration of gravity is 9.8 meters per second squared.

IV. (15) Set up and evaluate the integral to compute the arclength of the parabola y = 4x^{2}, 0 ≤ x ≤ 1. (Use the table to evaluate.)

V. All parts of this question refer to the differential equation

\frac{dy}{dx} = (y − 1)(y − 3)/4

A) (10) Sketch the slope field of this equation, showing the slopes at points on the lines y = 0, 1, 2, 3, 4, 5, and −3 ≤ x ≤ 3.

B) (5) On your slope field, sketch the graph of the solution of the equation with y(0) = 2.

C) (10) This is a separable equation; find an explicit formula for the solution satisfying the initial condition y(0) = 2.

Extra Credit (10) Set up the integrals to find the coordinates of the centroid of the region in the first quadrant inside the ellipse \frac{x^{2}}{4} + y^{2} = 1. You do not need to evaluate.