Question

Find the area bounded between the curves y²-1=2x and x=0.

Answer 1

y
$y ^{2} - 1 = 2x \\ y ^{2} - 1 = 2 \times 0 \\ y {}^{2} - 1 = 0$
y=±1

Answer 2

We need to find area bounded between,

• $x=\dfrac{y^2-1}{2}$

and,

• $x=0$

Assume,

$\longrightarrow \dfrac{y^2-1}{2}\leq0$

$\longrightarrow y^2\leq1$

$\longrightarrow y\in[-1,\ 1]$

Hence the area bounded will be,

$\displaystyle\longrightarrow A=\int\limits_{-1}^1\left(0-\dfrac{y^2-1}{2}\right)\ dy$

$\displaystyle\longrightarrow A=\int\limits_{-1}^1\left(\dfrac{1}{2}-\dfrac{y^2}{2}\right)\ dy$

$\displaystyle\longrightarrow A=\dfrac{1}{2}\big[y\big]_{-1}^1-\dfrac{1}{6}\left[y^3\right]_{-1}^1$

$\displaystyle\longrightarrow\underline{\underline{A=\dfrac{2}{3}}}$