Chapter 6, Problem 1RE

In Problems 1–3, set up definite integrals that represent the shaded areas in the figure over the indicated intervals.

1. Interval [a, b]

To determine

The definite integral that represents the indicated shaded area.

Answer to Problem 1RE

The definite integral ${\displaystyle {\int }_a^{b}f\left(x\right)dx}$ represents the indicated shaded area.

Explanation of Solution

Theorem used:

If f and g are continuous and $f\left(x\right)\ge g\left(x\right)$ over the interval $\left[a,b\right]$, then the area bounded by $y=f\left(x\right)$ and $y=g\left(x\right)$ for $a\le x\le b$ is given exactly by $A={\displaystyle {\int }_a^b\left[f\left(x\right)-g\left(x\right)\right]}dx$.

Calculation:

The given function is $f\left(x\right)$.

From the given figure, it is observed that the indicated area is bounded by the graph of the function $y=f\left(x\right)$ and $y=0$ over the interval $\left[a,b\right]$.

Also, it is noted that the function $f\left(x\right)$ does not have any discontinuity over the interval $\left[a,b\right]$. Thus, the function $f\left(x\right)$ is continuous and positive over the interval $\left[a,b\right]$.

By the above mentioned theorem, the area of the shaded region can be computed by the definite integral ${\displaystyle {\int }_a^{b}f\left(x\right)dx}$.

Thus, the definite integral ${\displaystyle {\int }_a^{b}f\left(x\right)dx}$ represents the indicated shaded area.