Suppose that the curve y = f(x) lies above the x-axis, so that f(x) ≥ 0 for all x in [a, b]. The area under the curve, that is, the area of the region bounded by the curve, the x-axis, and the lines x = a and x = b, equals

The definition of integral is made precisely in order to achieve this result.

When the function is positive

If f(x) ≤ 0 for all x in [a, b], the integral above is negative. However, it is still the case that its absolute value is equal to the area of the region bounded by the curve, the x-axis and the lines x = a and x = b. If y = f(x) crosses the x-axis, appropriate results hold. For example, if the regions A and B are as shown in the figure below, then

When the function changes sign

area of region $A={\int }_a^{b}f(x)\text{d}x$ and area of region $B={-\int }_b^{c}f(x)\text{d}x$.

It follows that

The combined areas of regions A and B is given by the integral ${\int }_a^b|f\left(x\right)|\text{d}x.$See signed area.

Polar areas

If a curve has an equation r = r(θ‎) in polar coordinates, there is an integral that gives the area of the region bounded by an arc AB of the curve and the two radial lines OA and OB. Suppose that ∠xOA = α‎ and ∠xOB = β‎. The area of the region described equals

$${\displaystyle {\int }_{\alpha }^{\beta }\frac{1}{2}{r}^2\text{d}\theta .}$$

Area of a polar sector