Can you plz solve this???
- Thread starter Rafid
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mmm4444bot
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We can help you solve it. Please follow the forum's guidelines and share how far you got. Do you have any specific questions?
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BigBeachBanana
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Hint: let [imath]u = f(u)[/imath].
Steven G
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Yes, I can solve it. The question is can you solve it? I suspect that since you came to this forum you need help solving this problem. What have you done? Where are you stuck?
Before solving an integral you need to know what you are integrating. Have you found f-1(u)?
Before solving an integral you need to know what you are integrating. Have you found f-1(u)?
Dr.Peterson
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This is a kind of problem that requires thought, not routine methods. So it is intended to get you thinking.
What thoughts have you had so far?
I would have started by sketching a graph of f(u), and thinking about what its inverse would look like. (Is it even invertible? Yes!) What area does this integral calculate?
Then I would realize that I wouldn't be able to find the inverse function analytically; in your case, you might only realize that it would be too hard for you. So I'd start thinking about how else I could find the same area on my graph ...
Look at the picture. You are asked to find the blue area.
blue area = rectangle area - green area
\(\displaystyle \int_{2\pi - 1}^{2\pi} f^{-1}(u) \ du = \ \)blue area \(\displaystyle \ = 2\pi - \int_{a}^{b} f(u) \ du\)
\(\displaystyle a\) and \(\displaystyle b\) are the \(\displaystyle u\) values when \(\displaystyle f(u)\) intersects the rectangle.
blue area = rectangle area - green area
\(\displaystyle \int_{2\pi - 1}^{2\pi} f^{-1}(u) \ du = \ \)blue area \(\displaystyle \ = 2\pi - \int_{a}^{b} f(u) \ du\)
\(\displaystyle a\) and \(\displaystyle b\) are the \(\displaystyle u\) values when \(\displaystyle f(u)\) intersects the rectangle.
Correction:
\(\displaystyle \int_{2\pi - 1}^{2\pi} f^{-1}(u) \ du = \ \)blue area \(\displaystyle \ = 2\pi - \left(\int_{a}^{b} f(u) \ du - (b - a)(2\pi - 1) \right)\)
Dr.Peterson
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Your thinking (after correction) agrees with mine. The trouble is, how do you find the value of your "a"? Are they expecting an exact value, or not?
Logically, a good approximation will be enough! If they expect an exact value, is it even possible to do that?