# Evaluate the improper iterated integral \int\limits_0^2 \int\limits_y^{2y} \left(10 + 5x^2 +...

## Question:

Evaluate the improper iterated integral

{eq}\int\limits_0^2 \int\limits_y^{2y} \left(10 + 5x^2 + 6y^2 \right) \text{d}x\text{d} y {/eq}

## Iterated Integral In Calculus:

This is a double Integration over a two-dimensional region on {eq}xy {/eq} space. The double integral can be written as: {eq}\displaystyle \int ^{d}_{c}\int ^{b}_{a}f\left( x,y\right) dxdy {/eq}

To solve this problem, we'll use the integral power rule, it is one of the most used rules that we can use to compute integration of a power function.

We are given:

{eq}\displaystyle \int\limits_0^2 \int\limits_y^{2y} \left(10 + 5x^2 + 6y^2 \right) \text{d}x\text{d} y {/eq}

Now, solving the innermost part of the integral, we will integrate in terms of {eq}x {/eq}:

{eq}= \displaystyle \int\limits_y^{2y} \left(10 + 5x^2 + 6y^2 \right) \text{d}x\\ {/eq}

Apply integral sum rule:

{eq}=\displaystyle \int\limits_y^{2y}10 \text{d}x + \int\limits_y^{2y} 5x^2 \text{d}x +\int\limits_y^{2y} 6y^2 \text{d}x {/eq}

Take the constant out:

{eq}=\displaystyle 10 \int\limits_y^{2y} \text{d}x +5 \int\limits_y^{2y} x^2 \text{d}x +6y^2 \int\limits_y^{2y} \text{d}x {/eq}

{eq}=\displaystyle \left[ 10x +\dfrac{5x^3}{3} +6y^2x \right]_y^{2y} {/eq}

{eq}=\displaystyle 20y +\dfrac{ 40 y^3}{3} +12y^3 - 10y -frac{5y^3}{3} -6y^3 {/eq}

{eq}=\displaystyle 10y +\dfrac{ 53 y^3}{3} {/eq}

Now, the integral will be:

{eq}\displaystyle \int_{0}^{2} (10y +\dfrac{ 53 y^3}{3} )\ dy \\ {/eq}

Take the constant out:

{eq}=\displaystyle 10 \int_{0}^{2} y \ dy + \int_{0}^{2} \dfrac{ 53 y^3}{3} \ dy {/eq}

{eq}=\displaystyle \left[ \dfrac{ 10 y^2 }{2}+ \dfrac{ 53 y^4}{12} \right]_{0}^{2} {/eq}

{eq}=\displaystyle 20+ \dfrac{ 212}{3} {/eq}

{eq}=\displaystyle \dfrac{ 272}{3} {/eq}

Therefore the solution is:

{eq}\displaystyle{\boxed{ \int\limits_0^2 \int\limits_y^{2y} \left(10 + 5x^2 + 6y^2 \right) \text{d}x\text{d} y = \dfrac{272}{3}}} {/eq}