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Vector Algebra

Question
CBSEENMA12032701
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Using definite integrals, find the area of the circle x2 + y2 = a2.

Solution

The equation of circle is
x2 + y2 = a2     ...(1)
Its centre is a and radius a. We know that circle x2 + y2 = a2 is symmetrical about both axes.
required area = 4 integral subscript 0 superscript straight a straight y space dx space equals space 4 space integral subscript 0 superscript straight a square root of straight a squared minus straight x squared end root dx
space equals space 4 space 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 space equals space 4 open square brackets open parentheses 0 plus straight a squared over 2 sin to the power of negative 1 end exponent 1 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 space equals space 4 open square brackets straight a squared over 2 cross times straight pi over 2 close square brackets space space space left square bracket because space sin to the power of negative 1 end exponent 0 space equals space 0 right square bracket equals space straight pi space straight a squared.

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