Area under a Curve

Area under a Curve

The area between the graph of y = f(x) and the x-axis is given by the definite integral below. This formula gives a positive result for a graph above the x-axis, and a negative result for a graph below the x-axis.

Note: If the graph of y = f(x) is partly above and partly below the x-axis, the formula given below generates the net area. That is, the area above the axis minus the area below the axis.

Formula:

Graph of y = f(x) above the x-axis with shaded yellow area between the curve and x-axis from x = a to x = b.

Example 1:

Find the area between y = 7 – x² and the x-axis between the values x = –1 and x = 2.

A semicircular curve from x=-2 to x=2 with maximum y=10, yellow shaded area under curve between x=0 and x=2.
Calculation showing Area = integral from -1 to 2 of (7 - x²)dx = (7x - x³/3) evaluated = 18

Example 2:

Find the net area between y = sin x and the x-axis between the values x = 0 and x = 2π.

A sine-like curve with x-axis labels 2, 4, 6; yellow shading above axis (0–~3.5) and below (~3.5–6).
Net area calculation: integral from 0 to 2π of sin x dx = (-cos x) from 0 to 2π = (-1)-(-1) = 0