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.

Area between Two Functions

Examples

1.Calculate the area of the space enclosed by the parabola y = x2 + 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:

Equation of the Line

Area between Two Functions

System of Equations

Integrals Solution

2.Find the area of the figure bounded by the function y = x2 and the lines y = x, x = 0 and x = 2

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

System of Equations

Area between Two Functions

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

Integrals

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

Integral

Total Area

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.

Intersection Point

Area of a Region

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.

Rectangular Area

The area under the curve y = ln x is:

Definite Integral

Derive

Integrate

Indefinite Integral

Integral

Total Area

4.Find the area of the enclosure limited by the parabola y = 4x − x2 and the tangents to the curve at the points of intersection with the x-axis.

Points of intersection:

Points of Intersection

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

Tangent

Tangent Equation

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

Slope

Tangent Equation

Area of a Region

Integrals

Integral Solution

5.Calculate the area bounded by the graphs of the functions y2 = 4x and y = x2.

System of Equations

Solution

Area between Two Functions

Integral Solution