Question

Evaluate the iterated integral. (give an exact answer.)int_0^4 int_x^(2x) int_0^y text( ) 2xyz text( ) dz text( ) dy text( ) dx

Answer 1

$\displaystyle\int_{x=0}^{x=4}\int_{y=x}^{y=2x}\int_{z=0}^{z=y}2xyz\,\mathrm dz\,\mathrm dy\,\mathrm dx$

$2xy\displaystyle\int_0^yz\,\mathrm dz=2xy\left[\frac{z^2}2\right]\bigg|_{z=0}^{z=y}=xy^3$

$x\displaystyle\int_x^{2x}y^3\,\mathrm dy=\frac{15}4x^5$

$\displaystyle\frac{15}4\int_0^4x^5\,\mathrm dx=\dfrac{15}{24}4^6=2560$