Find the area of region between parabolas $y^2 = 4ax$ and $x^2 = 4ay$.

Find the area of region between parabolas y² = 4ax and x² = 4ay.

Question 56 of 62

y^2 = 4ax

Maharashtra MsbshseMaharashtra HSC (MSBSHSE) Board 2022Subjective· 4mImportance★★★★★

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Find the intersection points, then integrate the difference of the two curves.

$y^2=4ax$ and $x^2=4ay$ . From the second, $y=\dfrac{x^2}{4a}$ ; substituting into the first:

$\left(\dfrac{x^2}{4a}\right)^2=4ax \implies \dfrac{x^4}{16a^2}=4ax \implies x^4=64a^3x \implies x(x^3-64a^3)=0$

So $x=0$ or $x=4a$ , giving intersection points $(0,0)$ and $(4a,4a)$ .

For $0\le x\le4a$ , $y=2\sqrt{ax}$ (upper branch of $y^2=4ax$ ) lies above $y=\dfrac{x^2}{4a}$ (checked e.g. at $x=a$ : $2a > a/4$ ).

$\text{Area} = \int_0^{4a}\left(2\sqrt{ax}-\dfrac{x^2}{4a}\right)dx = 2\sqrt a\left[\dfrac{x^{3/2}}{3/2}\right]_0^{4a} - \dfrac1{4a}\left[\dfrac{x^3}3\right]_0^{4a}$

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Q.Find the area of region between parabolas $y^2 = 4ax$ and $x^2 = 4ay$ .

Concept understanding — Area between Two Curves

Q.Find the area of region between parabolas y2=4axy^2 = 4axy2=4ax and x2=4ayx^2 = 4ayx2=4ay.

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Q.Find the area of region between parabolas $y^2 = 4ax$ and $x^2 = 4ay$ .

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