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=(15)/(24)4^6=2560`