Please help with a problem about definite integrals?

Sep 2018
27
0
Philippines
Hi everyone. I'm going through a problem on definite integrals at the moment (attached picture). I went online to try and find some sort of guide for the solution and managed to find something close enough. It is on this link: https://www.quora.com/How-do-you-solve-this-int-4-_-0-sqrt-16-x-2-dx

When I first tried to solve the problem, I did the usual method of substituting u and du. That didn't seem to work... In the link above, I see that they substituted x with 4sint instead. Why is that the case? Can someone please explain to me the reason behind this?

Thank you very much for your responses! (Happy)
 

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Oct 2009
1,245
555
Brisbane
Yes, this can be done using a trigonometric substitution. But maybe you haven't learnt that way yet.

I think this problem might be trying to get you to visualise the problem. Draw a graph and interpret the definite integral as the area under the curve.

The graph of \(\displaystyle y = \sqrt{16-x^2}\) is the top half of the circle with radius 4 and centre the origin. What is the area under the curve from x=-4 to x=4 (without using calculus)?
 
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