Draw a graph of $straight x squared over 9 plus straight y squared over 25 space equals space 1$ and evaluate area bounded by it.

The equation of ellipse is
$straight x squared over 9 plus straight y squared over 25 space equals space 1$
or $straight y squared over 25 space equals space 1 minus straight x squared over 9$
or $straight y squared space equals space 25 over 9 left parenthesis 9 minus straight x squared right parenthesis$

$therefore space space space space space straight y space equals space 5 over 3 square root of 9 minus straight x squared end root$ ...(1)
(in the first quadrant)
The ellipse is symmetrical about both the axes.
∴ required area = 4 (area AOB)
$equals space 4 integral subscript 0 superscript 3 straight y space dx space equals space 4 integral subscript 0 superscript 3 5 over 3 square root of 9 minus straight x squared end root space dx$ $open square brackets because space of space left parenthesis 1 right parenthesis close square brackets space space$
$equals space 20 over 3 integral square root of left parenthesis 3 right parenthesis squared minus straight x squared end root space dx space equals space 20 over 3 open square brackets fraction numerator straight x square root of 9 minus straight x squared end root over denominator 2 end fraction plus fraction numerator left parenthesis 3 right parenthesis squared over denominator 2 end fraction sin to the power of negative 1 end exponent open parentheses straight x over 3 close parentheses close square brackets subscript 0 superscript 3 equals space 20 over 3 open square brackets open parentheses fraction numerator 3 square root of 9 minus 9 end root over denominator 2 end fraction plus 9 over 2 sin to the power of negative 1 end exponent 1 close parentheses minus open parentheses 0 plus 9 over 2 sin to the power of negative 1 end exponent 0 close parentheses close square brackets equals space 20 over 3 open square brackets open parentheses 0 plus 9 over 2 cross times straight pi over 2 close parentheses minus left parenthesis 0 plus 0 right parenthesis close square brackets space equals space 20 over 3 cross times fraction numerator 9 straight pi over denominator 4 end fraction space equals space 15 space straight pi space sq. space units.$

121 Views

Application of Integrals

Hope you found this question and answer to be good. Find many more questions on Application of Integrals with answers for your assignments and practice.

Mathematics Part II

Browse through more topics from Mathematics Part II for questions and snapshot.

Draw a graph of and evaluate area bounded by it.

Application of Integrals

Mathematics Part II

Draw a graph of $straight x squared over 9 plus straight y squared over 25 space equals space 1$ and evaluate area bounded by it.