Area between a curve and the x-axis

To calculate the area between a curve and the \(x\)-axis we must evaluate using definite integrals.

Example

Calculate the shaded area below:

Area under the curve y=x^2-4x+5 between the y-axis and x=5

Solution

\[\int\limits_0^5 {({x^2}} - 4x + 5)dx\]

\[= \left[ {\frac{{{x^3}}}{3} - 2{x^2} + 5x} \right]_0^5\]

\[= \left( {\frac{{125}}{3} - 50 + 25} \right) - \left( 0 \right)\]

\[= \frac{{125}}{3} - 25\]

\[= \frac{{50}}{3}unit{s^2}\]

You may have to work out the limits of integration before calculating the area under a curve.

Example

Find the area under the curve \(y = 4x - {x^2}\)

Area between the curve y=4x-x^2 and the x-axis

Solution

First, we need to find out where the curve cuts the \(x\)-axis. Remember, a curve cuts the \(x\)-axis when \(y = 0\).

\[4x - {x^2} = 0\]

\(x(4 - x) = 0\) - factorise taking out a common factor of \(x\).

\(x = 0\) or \(4 - x = 0\)

Therefore \(x = 0\) or \(x = 4\)

The curve cuts the \(x\)-axis at \((0,0)\) and \((4,0)\)

\(0\) and \(4\) are the limits of integration.

Hence the area is:

\[\int\limits_0^4 {(4x - {x^2}} )dx\]

\[= \left[ {2{x^2} - \frac{{{x^3}}}{3}} \right]_0^4\]

\[= \left( {32 - \frac{{64}}{3}} \right) - \left( 0 \right)\]

\[= \frac{{32}}{3}unit{s^2}\]