Question

Find the value of integral \(\int_0^1\int_{x^2}^x xy(x+y)dydx\).

(a) ^3⁄15

(b) ^2⁄15

(c) ^2⁄30

(d) ^1⁄15

This question was addressed to me in an online quiz.

My doubt is from Application of Double Integrals in portion Multiple Integrals of Engineering Mathematics

Answer 1

Correct answer is (b) ^2⁄15

To explain I would say: Given, F(x)=\(\int_0^1\int_{x^2}^x xy(x+y)dydx=\int_0^1 \int_{x^2}^x(x ^2 y+xy^2)dydx\)

=\(\int_0^1 [\frac{x^2 y^2}{2}+\frac{xy^3}{3}] _x^{x^2}dx=\int_0^1 [\frac{x^3}{2}+\frac{x^4}{3}-\frac{x^4}{2}-\frac{x^5}{3}  ]dx=\frac{1}{2}+\frac{1}{3}-\frac{1}{2}-\frac{1}{5}=\frac{2}{15}\)