Question

x+sqrt(1-x^2)
How can I solve this by interpreting in terms of area?

Answer 1

That is the integrand from 0 to 1. Sorry

Answer 2

So the question is "find the integral?"
\[\int_0^1\left(x+\sqrt{1-x^2}\right)~dx\]

Answer 3

It asks to evaluate the integral by interpreting in terms of area.

Answer 4

I think I have an idea. Over the interval \([0,1]\), the area under the line \(x\) is a triangle, and the area under \(\sqrt{1-x^2}\) is a quarter-circle.
\[\begin{align*}\int_0^1\left(x+\sqrt{1-x^2}\right)~dx&=\int_0^1x~dx+\int_0^1\sqrt{1-x^2}~dx\\
&=\frac{1}{2}\text{(base)(height)}+\frac{1}{4}\pi\text{(radius)}^2\end{align*}\]