Chapter 5, Problem 2RCC

(a)

To determine

To write: The definition for the definite integral of a continuous function from a to b.

Explanation of Solution

Consider the function f which is continuous on interval $\left[a,b\right]$.

Divide the interval into n subintervals of equal width, $\Delta x=\frac{b-a}{n}$.

Let $x_0\left(=a\right),x_1,x_2,\mathrm{....},x_n\left(=b\right)$ be the end and intermediate points of the subintervals.

Then, the definite interval is the sum of the area of the subintervals. It is expressed as follows:

${\displaystyle {\int }_a^{b}f\left(x\right)dx}=\underset{n\to \infty }{\lim }{\displaystyle \sum _{i=1}^{n}f\left(x_i^{*}\right)\Delta x}$

Here, $x_i^{*}$ is any sample point in the $i^{th}$ subinterval $\left[x_{i-1},x_i\right]$.

(b)

To determine

To find: The geometric interpretation of ${\displaystyle {\int }_a^{b}f\left(x\right)dx}$ if $f\left(x\right)\ge 0$.

Explanation of Solution

The function $f\left(x\right)\ge 0$ represents the function in the first quadrant of the graph.

Sketch the curve $f\left(x\right)$ in the first quadrantas shown in Figure 1.

From Figure 1, the function f is positive when $f\ge 0$.

The geometric interpretation of ${\displaystyle {\int }_a^{b}f\left(x\right)dx}$ refers that the total area which is the region under the graph $y=f\left(x\right)$ and above the x-axis on the interval $\left[a,b\right]$.

(c)

To determine

To find: The geometric interpretation of ${\displaystyle {\int }_a^{b}f\left(x\right)dx}$, if $f\left(x\right)$ have both positive and negative values.

Explanation of Solution

The function $f\left(x\right)$ consists both positive and negative values which means that the graph of $y=f\left(x\right)$ forms the region above and below the x-axis.

The graph of function $y=f\left(x\right)$ for both positive and negative value of $f\left(x\right)$ is shown in the below Figure 1.

From the definition from part (a), as $f\left(x\right)$ takes both positive and negative values the limit of each Riemann sum is also positive, negative or zero.

Thus, the geometric interpretation of ${\displaystyle {\int }_a^{b}f\left(x\right)dx}$ refers the signed area under the graph $y=f\left(x\right)$ above the x-axis and below the x-axis on the interval $\left[a,b\right]$.