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  1. #1
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    Exclamation calc math question

    Hey,
    I was having a little trouble with these two calculus word problems. I would appreciate any help that I can get. Thanks a lot.

    1) Let R be the shaded region bounded by the graph of y = lnx and the line
    y = (x - 2).

    a) Find the area of R.

    b) Find the volume of the solid generated when R is rotated about the horizontal line y = -3.

    c) Write, but do not evaluate, an integral expression that can be used to find the volume of the solid generated when R is rotated about the y-axis.

    2) At an intersection in Thomasville, Oregon, cars turn left at the rate
    L(t) = 60 square root (t) sin^2(t/3) cars per hour over the time interval 0 <= t <= 18.

    a) To the nearest whole number, find the total number of cars turning left at the intersection over the time interval 0 <= t <= 18 hours.

    b) Traffic engineers will consider turn restrictions when L(t) >= 150 per hour. Find all values of t for which L(t) >= 150 and compute the average value of L over this time interval. Indicate units of measure.

    c) Traffic engineers will install a signal if there is any two-hour time interval during which the product of the total number of cars turning left and the total number of oncoming cars traveling straight through the intersection. Does this intersection require a traffic signal? Explain the reasoning that leads to your conclusion.
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  2. #2
    Eater of Worlds
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    For the solid of revolution, we can use washers or shells.

    Washers:

    {\pi}\int_{.158594}^{3.14619}\left[(ln(x)+3)^{2}-(x+1)^{2}\right]dx


    Shells:

    2{\pi}\int_{-1.841405}^{1.146193}(3+y)(y+2-e^{y})dy
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  3. #3
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    for the first part (finding the area), use \int_{0.158594}^{3.14619}\left[(lnx)-(x-2)\right]dx since observing that y=ln x is on top of y=x-2, we can integrate with respect to the x-axis. observing that x=0.158594 and x=3.14619 are the points where the two functions meet, we attain the limits of integration.
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  4. #4
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by Qcalc101 View Post
    Hey,
    I was having a little trouble with these two calculus word problems. I would appreciate any help that I can get. Thanks a lot.

    1) Let R be the shaded region bounded by the graph of y = lnx and the line
    y = (x - 2).

    a) Find the area of R.

    b) Find the volume of the solid generated when R is rotated about the horizontal line y = -3.

    c) Write, but do not evaluate, an integral expression that can be used to find the volume of the solid generated when R is rotated about the y-axis.

    2) At an intersection in Thomasville, Oregon, cars turn left at the rate
    L(t) = 60 square root (t) sin^2(t/3) cars per hour over the time interval 0 <= t <= 18.

    a) To the nearest whole number, find the total number of cars turning left at the intersection over the time interval 0 <= t <= 18 hours.

    b) Traffic engineers will consider turn restrictions when L(t) >= 150 per hour. Find all values of t for which L(t) >= 150 and compute the average value of L over this time interval. Indicate units of measure.

    c) Traffic engineers will install a signal if there is any two-hour time interval during which the product of the total number of cars turning left and the total number of oncoming cars traveling straight through the intersection. Does this intersection require a traffic signal? Explain the reasoning that leads to your conclusion.
    Ahh...the 2006 AP Calculus Free Response...I remember those...that was the year I took the exam...and passed...

    This may help you out, since I'm dead tired right now: 2006 AP Calculus AB Scoring Guidelines

    The explanations are vague, but its better than nothing!

    I'll answer #2 when I get the chance.
    Last edited by Chris L T521; May 29th 2008 at 12:29 AM.
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