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Math Help - Integrals - test tomorrow

  1. #1
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    Integrals - test tomorrow

    hey,

    i have a calc test tomorrow and i can't solve these practice problems. help please!!!

    Compute the following definite integrals:
    (a) integral from -1 to 1 of sin(x)sin(x^2)dx
    (b) integral from 0 to pi of cos(x) / (2 + sin(x)^2

    Given two curves y^2 - x = 0 and x + 2y = 3
    (a) Find the area of the region bounded between the curves
    (b) Find the centroid of the region
    (c) Find the volume of the solid obtained by revolving the region around the y-axis

    Consider a region with area equal to A lying in some plant. Consider a point P at a height h above the plane. Consider the solid 'cone' formed by joining P to all points of the region by line segments. Show that the volume of this solid is Ah/3. (Hint: Use the face that if you scale a region so that the distances get multiplied by a factor k, then the areas get scaled by the factor k^2.)

    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.

    Let f(x) = xe^(-x^2/2)
    (a) On what intervals does f increase? Decrease?
    (b) Find the extreme values of f.
    (c) Determine the concavity of the graph and find the points of inflection.
    (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?


    any help at all is very very very much appreciated!
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  2. #2
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    \int^1_{-1} sinxsin^2xdx

    \int^1_{-1} sinx \frac {1-cos2x} {2}

    \int^1_{-1} sinx-sinxcos2x

    \int^1_{-1} sinx- \frac {1} {2} (sin3x-sinx)

    \int^1_{-1} sinx- \frac {1} {2}sin3x+ \frac {1} {2}sinx

    \int^1_{-1} \frac {3} {2} sinx- \frac {1} {2}sin3x

    \frac{-3} {2}cosx+ \frac{1} {6}cos3x

    You can solve it from here.

    2) \int^{\pi}_0 \frac {cosx} {(2+sinx)^2}

    if you break the fraction up

    \int^{\pi}_0 \frac {cosx} {5} + \frac {1} {4tanx} - \frac {1} {cosx}

    BTW \int \frac {1} {tanx} = \ln|sinx|+C

    you can take it from here.
    Last edited by chengbin; March 15th 2009 at 09:06 AM.
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  3. #3
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    Quote Originally Posted by clue123 View Post
    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

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

    (b) Find the centroid of the region

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

    \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

    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.

    V = 2\pi \int_0^a x \cdot 2b\sqrt{1 - \frac{x^2}{a^2}} \, dx<br />

    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 ...

    x^2 + y^2 = 13^2

    A = \frac{1}{2}xy

    take the time derivative of the area function, substitute your known values, solve for dy/dt.
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  4. #4
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    Please note that urgent problems such as this should go in the Urgent Homework Section. If this was moved from that forum then I apologize.
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