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Calculus/Volume
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Question
Hi, I'm having some trouble with this:
"Let R be the region bounded by y=x, y=0, and x=2. Find the volume of the solid formed by revolving R about the line y=4."
Thanks!
"Let R be the region bounded by y=x, y=0, and x=2. Find the volume of the solid formed by revolving R about the line y=4."
What the Problem is Given:
The line y=x would be the the line through the origin at a 45° angle.
The line y = 0 is the x-axis.
The line x = 2 is the line that is straight up and down
and parallel to to the y-axis
Setting up the Integral
To revolve around the line y=4 means that we need to find the volume of rotating that triangle around the line y = 4.
That would be a double integral ∫∫ πr² dy dx where r is how far we are from y=4 That would make r = y - 4 with y going from x to 4 and x going from 0 to 2.
Thus, the problem is ∫∫π(y-4)² dy dx = π∫∫ y² - 8y + 16 dy dx.
Carrying it Out
Integrate with respect to y by addind one to the exponet in each number and dividing that number by the new exponet. Then evaluate that expression at at y = 4 and subtract off the expression at y = x.
Once this has been done, carry out the integration with respect to x in the same way, evaluating the new expression at x=4 and subtracting off the value of the expression at x=0.
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