For what value of 'a' is the area of the figure bounded by the lines, y = x 1 , y = 2 x − 1 1 , x = 2 \& x = a equal to ln 5 4 ?

Question

For what value of 'a' is the area of the figure bounded by the lines, y = x 1 ​ , y = 2 x − 1 1 ​ , x = 2 \& x = a equal to ln 5 ​ 4 ​ ?

TestHub · Accepted Answer

Use the antiderivative F ( x ) = ln x − 2 1 ln ( 2 x − 1 ) . Finite area requires a>1/2. Considering the intervals gives: a > 2 : F ( a ) − F ( 2 ) = ln 5 4 15 a 2 − 128 a + 64 = 0 a = 8 ( a > 2 ) . 2 1 < a < 1 : A = F ( a ) + F ( 2 ) 5 a 2 − 24 a + 12 = 0 a = 5 12 − 2 21 . Between 1 and 2 the area is at most ln(2/sqrt3), below the target. These are all solutions.

Mathematics - Area Under Curves Question with Solution | TestHub

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