18.01 Practice Exam 4

1. (10 points) Solve the differential equation \frac{dy}{dx} = y^{2}(x^{2} + 1).

2. (10 points) The population of a certain city satisfies the differential equation \frac{dx}{dt} = 0.05x where t is in years. How long does it take for the population to double?

3. (10 points) Find the area of the region between the curves y = 2x − x^{2}, y = x^{2}.

4. (10 points) Find the area under one arch of the curve y = 1 + sin(ax).

5. (10 points each) Find the area of the solid obtained by revolving around the x-axis the region bounded by the curves: y = x^{4}, y = 0, x = 1.

(a) using disks;

(b) using shells.

6. (10 points each) Consider the arc y = e^{x}, 0 ≤ x ≤ 1.

(a) Set up, but do not evaluate, the integral giving the length of the arc.

(b) Set up, but do not evaluate, the integral giving the area of the surface

obtained by revolving the arc about the x-axis.

7. (10 points each) Consider the region described in polar coordinates by 0 ≤ θ ≤ π/2, 0 ≤ r ≤ \sqrt{sin θ}.

(a) Find the area of the region.

(b) Find the length of the boundary of the region (set up, but do not evaluate, the integral involved).