2.6 Chapter 2 Review

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Sample Tests for Chapter 2

1 Sample Test 2a

(problem 1) Find the area between the curves $y = x^2 - 2$ and $y = 6 -x^2$ .

Points of intersection: $x = -2, 2$

$\answer {64/3}$

(problem 2) Find the volume of the solid obtained by revolving the region bounded by the graphs of $y = x/2$ and $y = \sqrt x$ about the $x$ -axis.

Points of intersection: $x = 0, 4$

Washer with $R = \sqrt x$ and $r = x/2$

$\answer {8\pi /3}$

(problem 3) Find the volume of the solid obtained by revolving the region bounded by the graphs of $y = x/2$ and $y = \sqrt x$ about the $y$ -axis.

Shell with $r = x$ and $h = \sqrt x - x/2$

$\answer {64\pi /15}$

(problem 4) Find the average value of $f(x) = \cos ^2(x)$ on the interval $[0, \pi /4]$ .

Half-angle formula: $\cos ^2(\theta ) = \frac 12 (1+\cos (2\theta ))$

$\answer {1/2 + 1/\pi }$

(problem 5) Find the length of the curve $\displaystyle y = \frac {x^2}{8} - \ln (x)$ over the interval $[1, 2]$ .

$L = \int _a^b \sqrt {1 + f'(x)^2} \, dx$

$\displaystyle f'(x)^2 =\left (\frac {x}{4} - \frac {1}{x}\right )^2 = \frac {x^2}{4} - \frac {1}{2} + \frac {1}{x^2}$

$\answer {3/8 + \ln (2)}$

(problem 6) Solve the initial value problem: $\displaystyle y' = \frac {y^2}{x},\, y(1) = 4$ .

Separate variables: $\frac {1}{y^2} dy = \frac {1}{x} dx$

$y = \answer {\frac {4}{1- 4\ln |x|}}$

(problem 7) Determine whether the series converges or diverges. If it converges, find its sum. $\displaystyle \sum _{n=2}^\infty \frac {3}{2^n}$
The series convergesdiverges and the sum is $S = \answer {3/2}$ .

(problem 8) Determine whether the series converges or diverges. If it converges, find its sum. $\displaystyle \sum _{n=1}^\infty \left (\frac {1}{n+1} - \frac {1}{n+3}\right )$
The series convergesdiverges and the sum is $S = \answer {5/6}$ .

2 Sample Test 2b

(problem 1) Find the area between the curves $x = y^2$ and $x = 2y + 3$ .

Integrate right curve minus left curve with respect to $y$

Points of intersection: $y = -1, 3$

$\answer {32/3}$

(problem 2) Find the volume of the solid obtained by revolving the region bounded by the graphs of $y = x-x^2$ and $y = 0$ about the line $y = 0$ .

Points of intersection: $x = 0, 1$

Disk with $R = x - x^2$

$\answer {\pi /30}$

(problem 3) Find the volume of the solid obtained by revolving the region bounded by the graphs of $y = x-x^2$ and $y = 0$ about the line $x = -1$ .

Shells $r = x+1$ and $h = x-x^2$

$\answer {\pi /2}$

(problem 4) Find the average value of $f(x) = xe^{3x}$ on the interval $[0, 2]$ .

Integration by Parts: $\int u \, dv = uv - \int v \, du$

$\answer {5e^6/18 + 1/18}$

(problem 5) Find the length of the curve $\displaystyle y = \ln (\sec (x))$ over the interval $[0, \pi /4]$ .

$f'(x) = \frac {1}{\sec (x)} \cdot \sec (x)\tan (x)$

$\int \sec (x) \, dx = \ln |\sec (x) + \tan (x)| + C$

$\answer {\ln (1 + \sqrt {2})}$

(problem 6) Solve the initial value problem: $\displaystyle y' = 4x^3 y,\, y(0) = 5$ .

Separate variables: $\frac {1}{y} dy = 4x^3 dx$

$y = \answer {5e^{x^4}}$

(problem 7) Determine whether the series converges or diverges. If it converges, find its sum. $\displaystyle \sum _{n=2}^\infty \frac {2^{2n}}{3^n}$
The series convergesdiverges and the sum is $S = \answer {\infty }$ .

(problem 8) Determine whether the series converges or diverges. If it converges, find its sum. $\displaystyle \sum _{n=1}^\infty \left (e^{\frac {1}{n}} - e^{\frac {1}{n+1}}\right )$
The series convergesdiverges and the sum is $S = \answer {e-1}$ .

2024-09-27 13:58:48