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.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 area of region A = ∫abf(x) dx and area of region B = − ∫bcf(x) dx.
It follows that Similarly, to find the area of the region bounded by a suitable curve, the y-axis, and lines y=c and y=d, an equation x=g(y) for the curve must be found. Then the required area equals
assuming that the curve is to the right of the y-axis, so that g(y) ≥ 0 for all y in [c, d]. As before, the value of the integral is negative if g(y)≤0.
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