Question

Find the area bounded by the curve y² = 4 a (x-1) and the lines x = 1 and y = 4 a.

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Solution

The equation of parabola is y² = 4 a (x-1) ...(1)
When y = 4 a , from (1) 16 a² = 4 a (x-1)
⇒ x - 1 = 4 a ⇒ x = 4 a + 1

therefore space space required space area space equals space integral subscript straight x space equals space 1 end subscript superscript straight x equals 4 straight a plus 1 end superscript straight y space dx space equals space integral subscript straight x equals 1 end subscript superscript straight x space equals space 4 straight a plus 1 end superscript 2 square root of straight a space square root of straight x minus 1 end root space dx

open square brackets because space of space left parenthesis 1 right parenthesis close square brackets

equals space 2 square root of straight a integral subscript 1 superscript 4 straight a plus 1 end superscript left parenthesis straight x minus 1 right parenthesis to the power of 1 half end exponent dx space equals space 2 square root of straight a open square brackets fraction numerator left parenthesis straight x minus 1 right parenthesis to the power of begin display style 3 over 2 end style end exponent over denominator begin display style 3 over 2 end style end fraction close square brackets subscript 1 superscript 4 straight a plus 1 end superscript equals space fraction numerator 4 square root of straight a over denominator 3 end fraction open square brackets left parenthesis straight x minus 1 right parenthesis to the power of 3 over 2 end exponent close square brackets subscript 1 superscript 4 straight a plus 1 end superscript space equals space fraction numerator 4 square root of straight a over denominator 3 end fraction open square brackets left parenthesis 4 straight a right parenthesis to the power of 3 over 2 end exponent minus 0 close square brackets space equals space fraction numerator 4 square root of straight a over denominator 3 end fraction cross times 8 straight a to the power of 3 over 2 end exponent equals space 32 over 3 space straight a squared space sq. space units

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