If a curve or a hybrid function looks like a familiar geometric shape when we graph it, we are easily able to find the area bounded by the curve(s) and the $x$x-axis.

Example 1

To find the shaded area on this graph we can simply calculate the area of a right triangle.

Area=$\frac{3\times6}{2}$3×62

Area=$9$9$units^2$units2

Example 2

To find the shaded area bounded by this curve and the x-axis we can find the area of a semicircle.

Area=$\frac{\pi\times4^2}{2}$π×422

Area=$25.13$25.13$units^2$units2

Example 3

How would we find the area bounded by the three lines and the $x$x-axis below?

We don't need to break it up into three different sections, although we could. Instead we can find the area of a trapezium.

Area=$\frac{2\left(2+5\right)}{2}$2(2+5)2

Area=$7$7$units^2$units2

Worked Examples

Question 1

Consider the function drawn below:

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Calculate geometrically, the area bounded by the curve and the $x$x-axis over $0\le x\le4$0≤x≤4.

Question 2

Find the exact value of $\int_0^{12}f\left(x\right)dx$∫120f(x)dx geometrically, where $y=f\left(x\right)$y=f(x) is graphed below.

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Question 3

Find the exact value of $\int_{-6}^6\sqrt{36-x^2}dx$∫6−6√36−x2dx geometrically.

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Question 4

The function $f\left(x\right)$f(x) is defined as:

$2x$2x

if $0\le x\le3$0≤x≤3

$f\left(x\right)$f(x)

$=$=

$6$6

if $33<x<6

$18-2x$18−2x

if $6\le x\le9$6≤x≤9

Graph $f\left(x\right)$f(x) on the axis below.

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Hence, calculate geometrically the area bounded by the curve and the $x$x-axis.