2 The area bounded by a curve lying above the x-axis

Consider the graph of the function $y=f\left(x\right)$ shown in Figure 8. Suppose we are interested in calculating the area underneath the graph and above the $x$ -axis, between the points where $x=a$ and $x=b$ . When such an area lies entirely above the $x$ -axis, as is clearly the case here, this area is given by the definite integral ${\int \; <!--nolimits\; -->}_a^{b}f\left(x\right)dx$ .

Figure 8

Example 12

Calculate the area bounded $y=x^{-1}$ and the $x$ -axis, between $x=1$ and $x=4$ .

Solution

Below is a graph of $y=x^{-1}$ . The area required is shaded; it lies entirely above the $x$ -axis.

Figure 9

Task!

Find the area bounded by the curve $y=sinx$ and the $x$ -axis between $x=0$ and $x=\pi$ . (The required area is shown in the figure. Note that it lies entirely above the $x$ -axis.)

Answer

Task!

Find the area under $f\left(x\right)={\text{e}}^{2x}$ from $x=1$ to $x=3$ given that the exponential function ${\text{e}}^{2x}$ is always positive.

Answer

The figure shows the graphs of $y=sinx$ and $y=cosx$ for $0\le x\le \frac{1}{2}\pi$ . The two graphs intersect at the point where $x=\frac{1}{4}\pi$ . Find the shaded area.

Figure 10

Solution

To find the shaded area we could calculate the area under the graph of $y=sinx$ for $x$ between 0 and $\frac{1}{4}\pi$ , and subtract this from the area under the graph of $y=cosx$ between the same limits. Alternatively the two processes can be combined into one and we can write

So the numeric value of the integral is $\frac{2}{\sqrt{2}}-1=0.414$ to 3 d.p.. (Alternatively you can use your calculator to obtain this result directly by evaluating $sin\frac{\pi }{4}$ and $cos\frac{\pi }{4}$ .)

Exercises

In each question you should check that the required area lies entirely above the horizontal axis.

1. Find the area under the curve $y=7x^2$ and above the $x$ -axis between $x=2$ and $x=5$ .
2. Find the area bounded by the curve $y=x^3$ and the $x$ -axis between $x=0$ and $x=2$ .
3. Find the area bounded by the curve $y=3t^2$ and the $t$ -axis between $t=-3$ and $t=3$ .
4. Find the area under $y=x^{-2}$ between $x=1$ and $x=10$ .

Answer