Area and volume applications
Beyond area under one curve, integrals find the area between two curves and volumes of solids. The area between y = x and y = x^2 on [0, 1] is the integral of (x - x^2), giving 1/2 - 1/3 = 1/6. Revolving the region under y = x on [0, 1] around the x-axis gives a cone with radius 1 and height 1: its volume is pi times the integral of x^2 dx, which works out to pi/3, matching the geometry formula for a cone's volume, one third times pi times r squared times h.
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