How to write area as integral?

**maiamorbific** · Sep 17th 2009, 08:33 PM

The question is:
Find the area of the region between y=sin(x) and y=x
for 0 ≤ x ≤ pi/2.
It's going to be a definite integral with bounds from 0 to pi/2, right? But what is the equation and how do I get it?
Thank you!

**redsoxfan325** · Sep 17th 2009, 09:23 PM

Originally Posted by maiamorbific

The question is:
Find the area of the region between y=sin(x) and y=x
for 0 ≤ x ≤ pi/2.
It's going to be a definite integral with bounds from 0 to pi/2, right? But what is the equation and how do I get it?
Thank you!

$x$ is greater than $\sin(x)$ for all $x>0$ , so you want to find the area under $y=x$ and subtract the area under $y=\sin(x)$ . The value you want is:

$\int_0^{\pi/2}x\,dx-\int_0^{\pi/2}\sin x\,dx$

This can be rewritten as a single integral: $\int_0^{\pi/2}(x-\sin x)\,dx$

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