Question

Find the value of \(\int_0^{1-y} xy\sqrt{1-x-y} \,dxdy\) where, y varies from 0 to 1.

(a) ^16⁄946

(b) ^8⁄945

(c) ^16⁄45

(d) ^16⁄945

This question was posed to me in unit test.

I'd like to ask this question from Application of Double Integrals in division Multiple Integrals of Engineering Mathematics

Answer 1

Correct answer is (d) ^16⁄945

Best explanation: Given, f(x)=\(\int_1^0∫_0^{1-y} xy\sqrt{1-x-y} \,dxdy\)

putting, \(t=\frac{x}{1-y}\)=>x=t(1-y)=>dx=(1-y)dt

\(\int_1^0\int_0^1 t(1-y)y\sqrt{1-t(1-y)-y} (1-y)dtdy\)

=\(\int_1^0\int_0^1 y(1-y)^{5/2} t(1-t)^{1/2} dtdy\)

=\(\int_1^0y(1-y)^{5/2} dy \int_0^1 t(1-t)^{1/2} dt\)

=\(\int_0^1 y^{2-1} (1-y)^{7/2-1} dy \int_0^1 t^{2-1} (1-t)^{3/2-1} dt=β(2,\frac{7}{2})β(2,\frac{3}{2})=\frac{16}{945}\)