Application of Definite Integration Questions with Solutions

This topic uses definite integrals to find the area of regions in the plane. It covers the area under a curve and above the x-axis, the area between a curve and the y-axis, the area between two curves, and the areas of standard regions bounded by circles, parabolas, ellipses and straight lines. Always start with a rough sketch of the region and find where the curves intersect, since those points become the limits of integration. For the area between two curves, integrate (upper curve - lower curve) between their intersection points. If part of a curve lies below the x-axis, take the absolute value of that part separately rather than integrating straight across, or the positive and negative areas will cancel. Useful standard results include the area of the ellipse x^2/a^2 + y^2/b^2 = 1, which is pi ab. Practise the area under the curve questions below, each with a sketch and a step-by-step solution.