Question

The function,f is defined by F(x,y)=x^2+xy+y^2 is integrable over [2,1]×[1,2]. True or false with full explanation

Answer 1

We have given the function,

$F(x,y)=x^2+xy+y^2$ over [2,1]×[1,2]

x varies from 2 to 1.

y varies from 1 to 2.

Integrating the given function,

$\int_{1}^{2} \int_{2}^{1} (x^2+xy+y^2)dxdy\\$

$\int_{1}^{2}[\dfrac{x^3}{3}+\dfrac{x^2y}{2}+xy^2]_{2}^{1}dy$

$= \int_{1}^{2}[-\dfrac{7}{3}+\dfrac{5y}{2}-y^2]dy$

$= [-\dfrac{7y}{3}+\dfrac{5y^2}{4}- \dfrac{y^3}{3}]_{1}^{2}$

$= -\dfrac{87}{12}$

Hence, the given function is Integrable over [2,1]×[1,2].