Question

Evaluate the iterated integral ∫(0 to 1)∫(x to 5x)∫(0 to y) 5xyz dzdydx.

Answer 1

$\int_0^1 \int_x^{5x} \int_0^y 5xyz dzdydx$

Go from inside to outside

$1. \int_{0}^{y} 5xyz dz = 5xy\Big[z^2/2\Big]_0^y = 5/2xy^3$

$2. \int_{x}^{5x} 5/2xy^3 dy = 5/2 x\Big[y^4/4\Big]_x^{5x} = 5/2x(625x^4/4 - x^4/4) = 390x^5$

$3. \int_0^1 390x^5 dx = 390x^6/6\Big|_0^1 = 65$

Answer: $\int_0^1 \int_x^{5x} \int_0^y 5xyz dzdydx = 65$