Find a constant so the area between two graphs is a specified constant

Let $c \in \mathbb{R}_{>0}$ be a positive real constant and define

$f(x) = x^2, \qquad g(x) = cx^3.$

The graphs of $f$ and $g$ intersect at the points $(0,0)$ and $\left( \frac{1}{c}, \frac{1}{c^2} \right)$ . Find a value for the constant $c$ so that the area between the graphs of $f$ and $g$ over the interval $left[ 0, \frac{1}{c} \right]$ has area $\frac{2}{3}$ .

First, for any $x \in \left[0, \frac{1}{c} \right]$ we have

$f(x) = x^2 = \frac{1}{c} \, cx^2 \geq cx^3 = g(x).$

Thus, the area between $f$ and $g$ is given by

$\begin{align*} \int_0^{\frac{1}{c}} (x^2 - cx^3) \, dx &= \int_0^{\frac{1}{c}} x^2 \, dx - c \int_0^{\frac{1}{c}} x^3 \, dx \\ &= \frac{1}{3c^3} - \frac{1}{4c^3} \\ &= \frac{1}{12c^3} \end{align*}$

Since we want the area to equal $\frac{2}{3}$ we then solve for $c$ ,

$\frac{1}{12c^3} = \frac{2}{3} \quad \implies \quad \frac{1}{c^3} = 8 \quad \implies \quad c = \frac{1}{2}.$

Stumbling Robot

Find a constant so the area between two graphs is a specified constant

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