How do you find the area between two lines?

First, find the x coordinates of where the lines intersect by setting the equations of the lines equal to each other. Then solve the quadratic (or polynomial) equation. Next, integrate both lines individually with the limits being the x coodinates of the intersection. Then subtract the area of the lower line from the area of the upper line to find the area between the two.

For example:

Find the area between y=4x and y=x² - 2x + 5.

To find the x coordinates, set them equal to each other and solve.

So... 4x = x² - 2x + 5 => 0 = x² - 6x + 5. Thus using either the quadratic formula or factorising, we find that these lines intesect when x=1 and x=5

Next, we need to integrate the lines between the limits found.

So... y=4x integrates to => [2x²] and when plugging in x=5, we get 50 and x=1 gives us 2. Thus the area of y=4x bound by the x axis, x=1 and x=5 is 50-2 = 48.

Similarly, y=x² - 2x + 5 integrates to => [x³/3 - x² + 5x]. Again, we put in x=5 and it gives 125/3 and x=1 gives us 13/3. Thus the area of y=x² - 2x + 5 bound by the x axis, x=1 and x=5 is 125/3 - 13/3 = 112/3.

We know by ploting the graphs that y=4x is above y=x² - 2x + 5. Hence, to find the area between these two lines is 48 - 112/3 = 32/3.