Area and Volume Formulas — Question 3

Question 3 - Solution

We approximate the region using vertical rectangles.

Partition the interval $[a,b]$ as $$a=x_0<x_1<\cdots<x_n=b,$$ and let $$\Delta x_i=x_i-x_{i-1}.$$

Choose a sample point $x_i^*$ in each subinterval $[x_{i-1},x_i]$.

At the point $x_i^*$, the height of the region between the curves is $$f(x_i^*)-g(x_i^*).$$

Thus, the area of the corresponding rectangle is $$\Delta A_i=\bigl(f(x_i^*)-g(x_i^*)\bigr)\Delta x_i.$$

The total area of all rectangles is approximated by the sum $$\sum_{i=1}^n \bigl(f(x_i^*)-g(x_i^*)\bigr)\Delta x_i.$$

As the partition is refined and $\|P\|\to 0$, these rectangles more accurately approximate the region between the curves. Since $f$ and $g$ are continuous, the limit of the sums exists.

Taking the limit gives $$A=\lim_{\|P\|\to 0}\sum_{i=1}^n \bigl(f(x_i^*)-g(x_i^*)\bigr)\Delta x_i =\int_a^b \bigl(f(x)-g(x)\bigr)\,dx.$$

$$\boxed{A=\int_a^b \bigl(f(x)-g(x)\bigr)\,dx}$$