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Question: If the area bounded by \(y = a x ^ { 2 }\)and \(x = a y ^ { 2 }\) , \(a > 0\) is 1, then a =...

If the area bounded by y=ax2y = a x ^ { 2 }and x=ay2x = a y ^ { 2 } , a>0a > 0 is 1, then a =

A

1

B

13\frac { 1 } { \sqrt { 3 } }

C

13\frac { 1 } { 3 }

D

−13- \frac { 1 } { \sqrt { 3 } }

Answer

13\frac { 1 } { \sqrt { 3 } }

Explanation

Solution

The x coordinate of A is 1a\frac { 1 } { a }

According to the given condition

1=∫01a(xa−ax2)dx=1a⋅23[x32]01a−a3[x3]01/a1 = \int _ { 0 } ^ { \frac { 1 } { a } } \left( \sqrt { \frac { x } { a } } - a x ^ { 2 } \right) d x = \frac { 1 } { \sqrt { a } } \cdot \frac { 2 } { 3 } \left[ x ^ { \frac { 3 } { 2 } } \right] _ { 0 } ^ { \frac { 1 } { a } } - \frac { a } { 3 } \left[ x ^ { 3 } \right] _ { 0 } ^ { 1 / a } ⇒a=13\Rightarrow a = \frac { 1 } { \sqrt { 3 } }