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I need to find the area between the graphs of $$f(x)=e^{x/2}-2 \quad\text{and}\quad g(x)=x^2+2x-1$$ and the lines $x = 0$ and $x = 2$.

I think I need to do $g(x)-f(x)$ with $0$ and $2$ as the lower and upper limits on the integrals.

But how do I go forward with it?

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  • $\begingroup$ Your idea is fine but the assumption $g(x) \ge f(x)$ needs first to be proved. Also we need to be sure we are looking for the absolute area and not the signed area. $\endgroup$ – user Nov 2 '20 at 10:14
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Yes your idea is fine, firstly we need to show that $g(x)\ge f(x)$ within the domain

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then the area between the curves is given by

$$\int_0^2 (g(x)-f(x))dx$$

which is the sum of trivial integrals.

Note that if we were interested in the signed area we need to divide the integral according to the different sign of the funtion in the domain of integration.

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