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Mth 174
Your 'question form' report has been received. Scroll down through the document to see any comments I might have inserted, and my final comment at the end.
** Question Form_labelMessages **
Integral of Fraction Question
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Find the indefinite integral of the function t^ 3 / e^( 8 t).
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I integrated this problem by parts. u = t^3, u` = 3t^2, v` = e^(-8t), v = -1/8 e^(-8t).
Then I used the formula, uv - integral of u' v which is:
t^3 * -1/8 e^(-8t) - integral of 3t^2 * -1/8 e^(-8t)
Then I took the parts of the integral of 3t^2 * -1/8 e^(-8t), but from there I got lost.
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You've got the first step, and the rest are no more difficult. You just need to keep track of you terms (and you will end up with 5 terms).
Note that
3t^2 * -1/8 e^(-8t) =
-3/8 t^2 e^(-8 t).
You can integrate t^2 e^(-8 t), then multiply the result by -3/8.
You'll use integration by parts again.
The integral part of what you'll get will involve t e^(-8 t), and will require one more integration by parts.
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#$&*
Mth 174
Your 'question form' report has been received. Scroll down through the document to see any comments I might have inserted, and my final comment at the end.
** Question Form_labelMessages **
Rate of Water Rising Question
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Problem Number 1
If water is rising at .74 cm/sec in a sphere of radius 47 cm then at what net rate in cm^3/sec is water entering the sphere when water depth is 30 cm?
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This is a question from the first test. I'm not even sure where to begin with this problem. I don't recall a question like this from the text.
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This is closely related to the geometry of the sphere.
You should be able to answer the following:
Using basic geometry you can figure out the cross-sectional area of the water surface at that depth.
That cross-sectional area will be changing, but it won't change appreciably in a very short time interval.
In a very short time interval `dt, assuming that the cross-sectional area doesn't change, what will be the added volume during that interval?
At what rate is the water therefore flowing into the sphere at the specified instant?
Note that you might find relevant information in some of the Class Notes.
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