Originally Posted by

**clue123** hey,

Compute the following definite integrals:

(a) integral from -1 to 1 of sin(x)sin(x^2)dx

hint ... the integrand is an odd function

(b) integral from 0 to pi of cos(x) / (2 + sin(x)^2

substitution, let u = sin(x) ... integral will involve an arctan

Given two curves y^2 - x = 0 and x + 2y = 3

(a) Find the area of the region bounded between the curves

$\displaystyle A = \int_{-3}^1 (3-2y) - y^2 \, dy$

(b) Find the centroid of the region

$\displaystyle \bar{x} = \frac{\int_{-3}^1 \frac{1}{2}[(3-2y)-y^2]^2 \, dy}{\int_{-3}^1 (3-2y) - y^2 \, dy}$

$\displaystyle \bar{y} = \frac{\int_{-3}^1 y[(3-2y)-y^2] \, dy}{\int_{-3}^1 (3-2y) - y^2 \, dy}$

(c) Find the volume of the solid obtained by revolving the region around the y-axis

$\displaystyle V = \pi \int_{-3}^1 (3-2y)^2 - (y^2)^2 \, dy$

Use the shell method to find the volume enclosed by the surface obtained by revolving the ellipse x^2/a^2 + y^2/b^2 = 1 about the y-axis.

$\displaystyle V = 2\pi \int_0^a x \cdot 2b\sqrt{1 - \frac{x^2}{a^2}} \, dx

$

Let f(x) = xe^(-x^2/2)

(a) On what intervals does f increase? Decrease?

f increases where f' > 0 ... decreases where f' < 0

(b) Find the extreme values of f.

determine extrema from where f' changes sign

(c) Determine the concavity of the graph and find the points of inflection.

concavity changes where f'' changes sign

(d) Sketch the graph indicating the asymptotes if any

A 13-foot ladder is leaning against a vertical wall. If the bottom of the ladder is being pulled away from the wall at the rate of 2 feet per second, how fast is the area of the triangle formed by the wall, the ground, and the ladder changing when the bottom of the ladder is 12 feet from the wall? How fast is the top of the ladder dropping?

relationship between the variables is ...

$\displaystyle x^2 + y^2 = 13^2$

$\displaystyle A = \frac{1}{2}xy$

take the time derivative of the area function, substitute your known values, solve for dy/dt.