Find the area between the two curves (parabolas) $y^2 = 7x$ and $x^2 = 7y$.

Find the area between the two curves (parabolas) y² = 7x and x² = 7y.

Question 14 of 19

y^2 = 7x

Maharashtra MsbshseMaharashtra HSC (MSBSHSE) Board 2022Subjective· 3mImportance★★★★★

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Find the intersection points $(0,0)$ and $(7,7)$ , then integrate $\big(\sqrt{7x}-\tfrac{x^2}{7}\big)$ from $0$ to $7$ to get $\dfrac{49}{3}$ sq. units.

Points of intersection. From $y^2=7x$ we get $y=\sqrt{7x}$ , and from $x^2=7y$ we get $y=\dfrac{x^2}{7}$ . Equating:

$\sqrt{7x}=\dfrac{x^2}{7}\ \Rightarrow\ 49\cdot 7x = x^4\ \Rightarrow\ x^4=343x\ \Rightarrow\ x(x^3-343)=0$ ,

so $x=0$ or $x=7$ . The curves meet at $(0,0)$ and $(7,7)$ .

Which curve is on top. For $0<x<7$ , the parabola $y=\sqrt{7x}$ lies above $y=\dfrac{x^2}{7}$ (e.g. at $x=1$ , $\sqrt7\approx 2.65 > \tfrac17$ ).

Set up and evaluate the area.

$A=\displaystyle\int_0^7\left(\sqrt{7x}-\dfrac{x^2}{7}\right)dx=\int_0^7\sqrt7\,x^{1/2}\,dx-\int_0^7\dfrac{x^2}{7}\,dx$ .

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Q.Find the area between the two curves (parabolas)
$y^2 = 7x$ and $x^2 = 7y$ .

Concept understanding — Area between Two Curves

Q.Find the area between the two curves (parabolas) y2=7xy^2 = 7xy2=7x and x2=7yx^2 = 7yx2=7y.

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Q.Find the area between the two curves (parabolas)
$y^2 = 7x$ and $x^2 = 7y$ .

Unlock everything free for 14 days