Area Between Curves

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The Area Between Two Curves

To get the area between two curves, $f$ and $g$, we slice the region between them into vertical strips, each of width $\Delta x$. Denote by $H(x)$ the height of the area at a point $x$. The area of each strip is roughly $H(x)\cdot \Delta x$.
Adding up the area strips, the total area is approximately $\displaystyle\sum_{i=1}^n H(x_i)\, \Delta x$. Taking a limit, the total area is exactly $\displaystyle\int_a^b H(x)\, dx$.

Let's look at an example. Suppose $f(x)\ge g(x)$ for all $x$ in the interval we are considering, as in the graph below, so that $H(x)=f(x)-g(x)$ on the interval $[0,10]$.

Then $$\displaystyle\int_a^b H(x)\, dx=\int_0^{10} (f(x)-g(x))\,dx=\int_0^{10}(2\cos x+8 -(3\sin x+4))\,dx=\int_0^{10}(2\cos x-3\sin x+4)\,dx.$$
DO: Before looking ahead, evalute the previous integral. Check your antiderivative!

$$\int_0^{10}(2\cos x-3\sin x+4)\,dx=(2\sin x+3\cos x+4x)\left|\begin{array}{c} ^{10} \\ _0 \end{array}\right .=2\sin(10)+3\cos(10)+4(10)-(2\cdot 0+3\cdot 1+4(0))$$$$=2\sin(10)+3\cos(10)+37\approx 2(-.544)+3(-.839)+37=33.395$$ (here we must round off the trig values, so our answer is no longer exact).

We look at more examples in the video below. Notice while watching that, as was true in the example above, you must know what the graphs of the two functions look like in order to know which function is the larger. Also notice that sometimes we are given the beginning and ending values of $x$ explicitly, and sometimes we have to figure out where two curves meet.