# Evaluate the iterated integral. \int_{0}^{2}\int_{0}^{6x^{2}}x^{3}dy dx

Evaluate the iterated integral.
${\int }_{0}^{2}{\int }_{0}^{6{x}^{2}}{x}^{3}dydx$
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Step 1
To evaluate the integral: ${\int }_{0}^{2}{\int }_{0}^{6{x}^{2}}{x}^{3}dydx$
${\int }_{0}^{2}{\int }_{0}^{6{x}^{2}}{x}^{3}dydx={\int }_{0}^{2}{x}^{3}\left[{\int }_{0}^{6{x}^{2}}dy\right]dx$
$={\int }_{0}^{2}{x}^{3}{\left[y\right]}_{0}^{6{x}^{2}}dx$
$={\int }_{0}^{2}{x}^{3}\left[6{x}^{2}\right]dx$
$=6{\int }_{0}^{2}{x}^{5}dx$
$=6{\left[\frac{{x}^{6}}{6}\right]}_{0}^{2}$
$=6\left[\frac{{2}^{6}}{6}-0\right]$
$={2}^{6}$
=64
Step 2