Question

Using definite integrals, find the area of the ellipse $straight x squared over straight a squared plus straight y squared over straight b squared equals 1$ .

Solution

The equation of the ellipse is
$straight x squared over straight a squared plus straight y squared over straight b squared space equals space 1$ ...(1)
The ellipse is symmetrical about the axes.
∴ required area = 4 (area OAB)
$equals space 4 integral subscript 0 superscript straight a straight y space dx space equals space 4 integral subscript 0 superscript straight a straight b over straight a square root of straight a squared minus straight x squared end root space dx$ [ $because space of space left parenthesis 1 right parenthesis$ ]
$equals space 4 straight b over straight a integral subscript 0 superscript straight a square root of straight a squared minus straight x squared end root dx$
$equals 4 straight b over straight a open square brackets fraction numerator straight x square root of straight a squared minus straight x squared end root over denominator 2 end fraction plus straight a squared over 2 sin to the power of negative 1 end exponent straight x over straight a close square brackets subscript 0 superscript straight a equals space fraction numerator 4 straight b over denominator straight a end fraction open square brackets open parentheses fraction numerator straight a square root of straight a squared minus straight a squared end root over denominator 2 end fraction plus straight a squared over 2 sin to the power of negative 1 end exponent straight a over straight a close parentheses minus open parentheses 0 plus straight a squared over 2 sin to the power of negative 1 end exponent 0 close parentheses close square brackets equals space 4 straight b over straight a open square brackets open parentheses 0 plus straight a squared over 2. straight pi over 2 close parentheses minus left parenthesis 0 plus 0 right parenthesis close square brackets space equals space straight pi space straight a space straight b space sq. space units$

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Vector Algebra

Using definite integrals, find the area of the ellipse $straight x squared over straight a squared plus straight y squared over straight b squared equals 1$ .

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For Daily Free Study Material Join wiredfaculty

Vector Algebra

Using definite integrals, find the area of the ellipse .

Some More Questions From Vector Algebra Chapter

Mock Test Series

NCERT Sample Papers

Entrance Exams Preparation

Using definite integrals, find the area of the ellipse $straight x squared over straight a squared plus straight y squared over straight b squared equals 1$ .