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Numerade Educator

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Problem 78 Hard Difficulty

Evaluate $ \displaystyle \int^1_0 x \sqrt{1 - x^4} \,dx $ by making a substitution and interpreting the resulting integral in terms of an area.

Answer

$\frac{1}{8} \pi$

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Video Transcript

okay for this interval, let us say that you is ax squared to get d'you is two acts deep cracks, which means you can pull up the constant of 1/2. We can rewrite this in terms of argue. That's the whole point of doing use of petition is it makes it easier to integrate because we can report it and were easy to fall away. This is equivalent to 1/2 off the area 1/2 times area. Okay, now in the image gets seen. The shaded region was looked like this approximately 1/4 of a circle, which means 1/4 of a circle of radius one is one fourth times pi, which means 1/2 times area is 1/2 times 1/4 pie, which is 1/8 pie, which is our solution.