Find the area of the region bounded by the ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$, by integration.

> [!TLDR] > Use symmetry to write the ellipse's area as 4 times the first-quadrant area, then integrate using the standard $\sqrt{a^2-x^2}$ formula. > > 1. **Solve for $y$.** From $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$, > $$y = \frac{b}{a}\sqrt{a^2-x^2} \quad (\text{taking the upper half}).$$ > …

Find the area of the region bounded by the ellipse

Question 62 of 96

\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1

Tamil Nadu DgeTamil Nadu HSC (DGE) Board 2017Subjective· 10mImportance★★★★★est

65% · 62/96 Questions

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By symmetry, Area $=4\displaystyle\int_0^a y\,dx$ with $y=\dfrac{b}{a}\sqrt{a^2-x^2 …

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Question 62 of 96

\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1

Tamil Nadu DgeTamil Nadu HSC (DGE) Board 2017Subjective· 10mImportance★★★★★est

65% · 62/96 Questions

You're viewing a preview — the full solution, concept, methods & PYQ mapping are locked.

By symmetry, Area $=4\displaystyle\int_0^a y\,dx$ with $y=\dfrac{b}{a}\sqrt{a^2-x^2 …

Q.Find the area of the region bounded by the ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ , by integration.

Q.Find the area of the region bounded by the ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ , by integration.

Q.Find the area of the region bounded by the ellipse x2a2+y2b2=1\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1a2x2​+b2y2​=1, by integration.

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Q.Find the area of the region bounded by the ellipse x2a2+y2b2=1\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1a2x2​+b2y2​=1, by integration.

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Q.Find the area of the region bounded by the ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ , by integration.

Unlock the full solution

Q.Find the area of the region bounded by the ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ , by integration.

Unlock the full solution