y = x, y= x^2 find the area bounded by these curved and the ordinates x = 0, x = 3.

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Borys Shumyatskiy | College Teacher | (Level 3) Associate Educator

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Denote y1=x, y2=x^2. The area in question is the integral dx from x=0 to x=3 of |y1-y2|. It is evident that for x from 0 to 1 y1>=y2 and for x from 1 to 3 y2>=y1. So the area is int_0_1_(x-x^2)dx + int_1_3_(x^2-x)dx = ((1/2)x^2-(1/3)x_3)_0_1 + ((1/3)x^3-(1/2)x^2)_1_3 = (1/2-1/3) + (9-9/2-(1/3-1/2)) = 1/6 + 4.5 + 1/6 = 4.5 + 1/3 = 4 and 5/6.