## Quick Reference

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

*∫*_{a}^{b}*f*(*x*) *dx*.

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* = *∫*_{a}^{b}*f*(*x*) *dx* and area of region *B* = − *∫*_{b}^{c}*f*(*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

*∫*_{c}^{d}*g*(*y*) *dy*,

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.

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

*Subjects:*
Mathematics.