# Area between Two Functions

The area between two functions is equal to the area of the function located above minus the area of the function that lies below.

#### Examples

1.Calculate the area of the space enclosed by the parabola y = x^{2} + 2 and the straight line that passes through the points A(−1, 0) and B(1, 4).

The equation of the straight line that passing through AB:

2.Find the area of the figure bounded by the function y = x^{2} and the lines y = x, x = 0 and x = 2

The points of intersection of the parabola and the straight line y = x.

From x = 0 to x = 1, the straight line is above the parabola.

From x = 1 x = 2, the straight line is below the parabola.

3.Find the area of the plane region bounded by y = ln x, y = 2 and the coordinate axes.

Calculate the intersection point of the curve and the line y = 2.

The area is equal to the area of the rectangle ABC0 minus the area under the curve y = ln x.

The rectangular area is the base times height.

The area under the curve y = ln x is:

4.Find the area of the enclosure limited by the parabola y = 4x − x^{2} and the tangents to the curve at the points of intersection with the x-axis.

Points of intersection:

The equation of the tangent to the parabola at the point (0, 0):

Equation of the tangent to the parabola at the point (4, 0):

5.Calculate the area bounded by the graphs of the functions y^{2} = 4x and y = x^{2}.