Using integration, we can find the distance given the velocity. Definite integrals form the powerful tool to find the area under simple curves, the area bounded by a curve and a line, the area between two curves, the volume of the solids. The displacement and motion problems also find their applications of integrals. The area of the region enclosed between two curves y = f(x) and y = g(x) and the lines x =a, x =b is given by
Area = b∫a(f(x)−g(x))dx∫ab(f(x)−g(x))dx
Let us find the area bounded by the curve y = x and y = x2 that intersect at (0,0)and (1,1).
The given curves are that of a line and a parabola. The area bounded by the curves = 1∫0(y2−y1)dx∫01(y2−y1)dx
Area = 1∫0(x−x2)dx∫01(x−x2)dx
= x2 /2- x 3/3
= 1/2-1/3
= 1/6 sq units.