Area and Volume Formulas — Question 1

Question 1 - Solution

We interpret area using approximating rectangles.

Let $$a=x_0<x_1<\cdots<x_n=b$$ be a partition of $[a,b]$, and let $$\Delta x_i=x_i-x_{i-1}.$$

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

Since $f(x)\ge 0$, the area of the rectangle over $[x_{i-1},x_i]$ with height $f(x_i^*)$ is $$\text{Area}_i=f(x_i^*)\,\Delta x_i.$$

The total area of all rectangles is $$\sum_{i=1}^n f(x_i^*)\,\Delta x_i.$$

As the partition is refined (that is, as $\|P\|\to 0$), the rectangles better approximate the region under the curve. Because $f$ is continuous on $[a,b]$, it is integrable, and the limit of these sums exists.

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

By definition, this limit equals the exact area of the region under the curve.

Therefore, the area of the region bounded by $y=f(x)$, the $x$-axis, and $x=a$, $x=b$ is $$A=\int_a^b f(x)\,dx.$$

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