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Question

# On the same set of axes, draw the graphs of the relations $y=2 x \text { and } y^{2}=4 x.$ Determine the area of the region enclosed by these relations.

Solution

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Graph is shown below. The line $y=2x$ crosses $y^{2}=4x$ when

\begin{align*} 2x&=\sqrt{4x}\\ 4x^{2} &= 4x \\ 4x^{2} -4x &= 0\\ x\left(x-1\right)&=0 \end{align*}

$x=0$ and $x=1$. Therefore,

\begin{align*} \text{Area} &= \int_{0}^{1} \left[2\sqrt{x} - 2x \right]dx\\ &= \bigg[ \dfrac{4}{3}x^{\frac{3}{2}}-x^{2} \bigg]_{0}^{1}\\ &=\dfrac{4}{3}-1\\ &=\dfrac{1}{3} \; \text{units}^{2} \end{align*}

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