REVIEW, MIDTERM #1: MATH 1B

Problem 1. The integral

∫_{a}^{∞} \frac{1}{x^{\sqrt{3}}} dx

converges:

(a) For all values of a > 0;

(b) For all values of a ≥ 0;

(c) For a = 1 only;

(d) For no value of a.

Problem 2. Evaluate the integral

∫ \frac{4x^{2} + 4x + 1}{4x^{2} − 4x + 1} dx.

REVIEW, MIDTERM #1: MATH 1B

Problem 3. The area of the surface obtained by rotating the curve y = f(x) from a ≤ x ≤ b about the x-axis is given by:

(a) ∫_{a}^{b} πf(y)^{2} dy;

(b) ∫_{a}^{b} 2πy\sqrt{1 + (\frac{df}{dx})^{2}} dx;

(c) ∫_{f(a)}^{f(b)} 2πx\sqrt{1 + (\frac{df}{dx})^{2}} dy.

(d) ∫_{f(a)}^{f(b)} 2πy\sqrt{1 + (\frac{df}{dx})^{2}} dx.

Problem 4. Evaluate the integral ∫ \sqrt{x} ln x dx.

REVIEW, MIDTERM #1: MATH 1B

Problem 5. Which of the following does not have an elementary antiderivative?

(a) x^{2} ln x;
(b) sinh x;
(c) xe^{x^{2}};
(d) e^{x^{2}}.

Problem 6. Evaluate the integral ∫ csc^{4} x dx.

REVIEW, MIDTERM #1: MATH 1B

Problem 7. If the function f(x) is continuous on −1 ≤ x ≤ 1 except at the point x = a for some −1 < a < 1, then the improper integral

∫_{−1}^{1} f(x) dx

is written:

(a) lim_{t→a^{+}} ∫_{−1}^{t} f(x) dx + lim_{t→a^{−}} ∫_{t}^{1} f(x) dx;

(b) lim_{x→a} ∫_{−1}^{x} f(x) dx;
(c) (df/dx)(a) − lim_{t→a} ∫_{−t}^{t} f(x) dx;
(d) lim_{a→∞} ∫_{−1}^{1} f(x) dx.

Problem 8. Evaluate the integral ∫ \sqrt{x^{2} − 1} dx.

REVIEW, MIDTERM #1: MATH 1B

Problem 9. The sequence

a_{n} = \frac{2n^{2}}{n^{2} + 1}

is:

(a) Bounded, monotonic, and convergent;

(b) Unbounded, monotonic, and not convergent;

(c) Bounded, not monotonic, and convergent;

(d) Bounded, not monotonic, and not convergent.

Problem 10. Find the area of the surface obtained by rotating the curve

y = x^{2} + 1

about the y-axis for 0 ≤ x ≤ 1.

REVIEW, MIDTERM #1: MATH 1B

Problem 11. The partial fractions decomposition of

\frac{1}{(x − 3)^{2}(x^{2} + 3)}

is:

(a) \frac{A}{x − 3} + \frac{Bx + C}{(x − 3)^{2}} + \frac{Dx + E}{x^{2} + 3} ;

(b) \frac{A}{(x − 3)^{2}} + \frac{B}{x^{2} + 3};

(c) \frac{A}{x − 3} + \frac{B}{(x − 3)^{2}} + \frac{Cx + D}{x^{2} + 3} ;

(d) \frac{Ax + B}{(x − 3)^{2}} + \frac{Cx + D}{x^{2} + 3} ;

(e) \frac{A}{x − 3} + \frac{B}{(x − 3)^{2}} + \frac{C}{x^{2} + 3}.

Problem 12. Evaluate the integral ∫_{0}^{π} sin^{5} x cos^{2} x dx.

REVIEW, MIDTERM #1: MATH 1B

Problem 13. For the integral

∫_{0}^{2} e^{x} dx,

given the formula

|E_{S}| ≤ \frac{K(b − a)^{5}}{180n^{4}}

with |f^{(4)}(x)| ≤ K for a ≤ x ≤ b, the error in approximating the integral using Simpson’s rule for n = 2 is:

(a) |E_{S}| ≤ 1/90;

(b) |E_{S}| ≤ e/90;
(c) |E_{S}| ≤ e^{2}/90;
(d) |E_{S}| ≤ e^{2}/2880.

Problem 14. Determine if the integral

∫_{1}^{2} \frac{1}{x\sqrt{ln x}} dx

is convergent, and evaluate it if so.

REVIEW, MIDTERM #1: MATH 1B

Problem 15. The sum of the series

1 − \frac{1}{2} + \frac{1}{4} − \frac{1}{8} + …

is:

(a) 1/2;

(b) 1;

(c) 2/3;

(d) 3/2;

(e) The series is divergent.

Problem 16. Evaluate

∫_{e}^{e^{2}} \frac{ln(ln x)}{x} dx.

REVIEW, MIDTERM #1: MATH 1B

Problem 17. For the integral

∫_{a}^{b}(2x^{2} + 3x − 4) dx,

which of the following gives the best approximation for fixed n?

(a) Left endpoint approximation;

(b) Right endpoint approximation;

(c) Midpoint approximation;

(d) The Trapezoidal rule;

(e) Simpson’s rule.

Problem 18. Set up an integral to compute the length of the curve y = e^{x} for 1 ≤ y ≤ e. Suggest a substitution, but do not evaluate completely.

REVIEW, MIDTERM #1: MATH 1B

Problem 19. If f is continuous, then

∫_{−∞}^{∞} f(x) dx = lim_{t→∞} ∫_{−t}^{t} f(x) dx.

(a) True;

(b) False;

(c) Cannot be determined from the information given.

Problem 20. Determine if the integral

∫_{1}^{∞} \begin{matrix}√ & \overline{1 + \sqrt{x}} \\ √ & \overline{x} \end{matrix} dx

is convergent, and evaluate it if so.