MATH 21B

Final | Fall '16
1. (20 points) Compute the following indefinite integrals:
a) $\int \sin(2x + 3) dx$

b) $\int \frac{x \; dx}{\sqrt{x^2 + 1}} dx$

c) $\int \frac{dx}{(x + 1)(x + 2)(x + 3)} dx$

d) $\int \frac{\sqrt{1 - x^2}}{x^2} dx$

2. (20 points) Compute the following definite integrals:
a) $\int_0^1 \sqrt{x} \cdot (x + 1) dx$

b) $\int_1^2 \ln x \; dx$

c) $\int_0^\pi \sin^3 t \; dt$

d) $\int_0^1 \frac{2x + 1}{x^2 + 1} dx$

3. (10 points) Solve the differential equation $y' = \frac{1}{xy}$.

4. (10 points) Determine if the improper integral $\int_0^{+ \infty} \frac{dx}{x^2 + 4x + 5}$ converges or diverges. If it converges, compute its value.

5. (20 points) Consider the region $R$ bounded by the curves $y = e^x, y = 0, x = 0 \; \text{and} \; x = 1$ a) Find the area of $R$.

b) Find the volume of the solid of revolution obtained by rotation of $R$ about the $x$-axis.

c) Find the volume of the solid of revolution obtained by rotation of $R$ about the $y$-axis.

d) Find the coordinates of the center of mass of $R$.

6. (10 points) Compute the length of the curve defined by the equations: $x(t) = \cos t + \sin t, y(t) = \cos t - \sin t, 0 \le t \le 2\pi$
7. (10 points) A fish tank has the shape of parallelepiped (all faces are rectangles) with height $a$, width $b$ and length $c$. It is full of water. Compute the total force of water pressure on each of its faces (4 sides, top and bottom), assuming that the water density equals ρ and the gravity acceleration equals $g$.
8. (10 points) (Bonus problem) Find a function $f(x)$ such that $\int f^2(x) dx = \left( \int f(x) dx \right)^2$ Hint: define $F(x) = \int f(x) dx$ and write a differential equation for $F(x)$.