Question

Find the area of the region bounded by the curve ay² = x³, the y-axis and the lines y = a and y = 2a.

Answer 1

We have

Area BMNC = `int_"a"^(2"a") x"d"y`

= `int_"a"^(2"a") "a"^(1/3) y^(2/3) "d"y`

= `(3"a"^(1/3))/5|y^(5/3)|_"a"^(2"a")`

= `(3"a"^(1/3))/5|(2"a")^(5/3) - "a"^(5/3)|`

= `3/5 "a"^(1/3) "a"^(5/3) |(2)^(5/3) - 1|`

= `3/5 "a"^2 |2.2^(2/3) - 1|` sq.units