Question

Find the integral of f(x,y ) x^4+y^2 over the region bounded by y=x , y= 2x and x=2

Answer 1

$S: ( y=x, y = 2x, x = 2 )$

$\iint x^4+y^2 dS = \\ S \\ \int_0^2 \int_x^{2x} x^4 + y^2 dydx =$

$\int_0^2 \big[ x^4y + \cfrac13 y^3 \big]_x^{2x} dx =$

$\int_0^2 x^4(2x-x)+\cfrac13(8x^3-x^3) dx =$

$\int_0^2 x^5+\cfrac73x^3 dx = \big[ \cfrac16x^6 + \cfrac7{12}x^4 \big]_0^2 =$

$\cfrac16(2^6 - 0^6) + \cfrac7{12}(2^4 - 0^4) = \cfrac{2^2(2^3 + 7)}3 = 20$