Double Integrals

Calculus Level 2

Evaluate the following integral

$\large\int_{1}^{2}\int_{0}^{2y}( x^2+y^2)\ dx\ dy$

\dfrac{14}{3}

\dfrac{210}{2}

\dfrac{15}{4}

\dfrac{35}{2}

1 solution

A Former Brilliant Member
May 5, 2017

$\color{#3D99F6}\large \int_{1}^{2}\int_{0}^{2y}( x^2+y^2)dxdy$

$\color{#3D99F6}\large=\int_{1}^{2}\left[\dfrac{x^3}{3}+y^2x \right]_{0}^{2y}dy$

$\color{#3D99F6}\large=\int_{1}^{2}\left[\dfrac{(2y)^3}{3}+y^2(2y) \right]dy$

$\color{#3D99F6}\large=\int_{1}^{2}\left[\dfrac{8y^3}{3}+2y^3\right]dy$

$\color{#3D99F6}\large=\dfrac{14}{3}\int_{1}^{2}y^3dy$

$\color{#3D99F6}\large=\dfrac{14}{3}\left[\dfrac{y^4}{4}\right]_{1}^{2}$

$\color{#3D99F6}\large=\dfrac{14}{3}\left[\dfrac{2^4}{4}-\dfrac{1^4}{4}\right]$

$\color{#3D99F6}\large=\dfrac{14}{3}\left(\dfrac{15}{4}\right)$

$\color{#3D99F6}\large=\dfrac{210}{12}$

$\color{#3D99F6}\large=\dfrac{35}{2}$