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Compute the area between the graphs of functions

Let

    \[ f(x) = x^3 + x^2, \qquad g(x) = x^3+1. \]

Find the area between the graphs of f and g on the interval [-1,1].


First, we draw the graph, shading the region S between the two graphs in blue.

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Then, we calculate the area, a(S) of S:

    \begin{align*}   a(S) = \int_{-1}^1 (g(x) - f(x)) \, dx &= \int_{-1}^1 ((x^3+1)-(x^3+x^2)) \, dx\\   &= \int_{-1}^1 (1-x^2) \, dx \\   &= \left.x \biggr\rvert_{-1}^1 - \left. \frac{x^3}{3} \right|_{-1}^1 \\   &= (1+1) - \left( \frac{1}{3} + \frac{1}{3} \right) \\   &= \frac{4}{3}. \end{align*}

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