Section 7.1 Areas
Our study of integrals was motivated by the area problem so it feels necessary to at least note that in a fairly broad sense, we’ve solved the area problem. Consider, for example, \(f(x)=\sqrt{x}\) and \(g(x)=x^3\text{.}\) Let’s graph them both together. 
Take a look at that oddly shaped area trapped between the two curves. Can we calculate that area? Sure. We have the skills now. Notice that the area occurs over the interval \(0\le x \le 1\text{.}\) The curve on top is \(f(x)=\sqrt{x}\text{,}\) 
An antiderivative of \(\sqrt{x}\) is \(2x^{3/2}/3\) so that according to the FTC, the area under that curve is \(\int_0^1 \sqrt{x} dx = 2(1)^{3/2}/3 - 2(0)^{3/2}/3 = 2/3\text{.}\) The curve on the bottom is \(g(x)=x^3\text{,}\) 
An antiderivative of \(x^3\) is \(x^4/4\) so that according to the FTC, the area under that curve is \(\int_0^1 x^3 dx = (1)^{4}/4 - (0)^{4}/4 = 1/4\text{.}\) So, the area trapped between is \(2/3-1/4 = 8/12 - 3/12 = 5/12\text{.}\) -- very quick to calculutate with the skills we now have.
In general, it is apparent that if the graph of some function, say \(f(x)\text{,}\) is above the graph of some other function, say \(g(x)\text{,}\) over an interval of \(x\text{,}\) say \([a,b]\) then the area trapped between them can be calculated as
\begin{equation*}
\begin{array}{lll}
\int_a^b f(x)dx - \int_a^b g(x)dx \amp = \amp \int_a^b (f(x)-g(x))dx
\end{array}
\end{equation*}
