Answer to Question #104514 in Calculus for maria

Question #104514
Find the integral of f(x,y ) x^4+y^2 over the region bounded by y=x , y= 2x and x=2
1
Expert's answer
2020-03-06T12:27:33-0500

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


x4+y2dS=S02x2xx4+y2dydx=\iint x^4+y^2 dS = \\ S \\ \int_0^2 \int_x^{2x} x^4 + y^2 dydx =


02[x4y+13y3]x2xdx=\int_0^2 \big[ x^4y + \cfrac13 y^3 \big]_x^{2x} dx =


02x4(2xx)+13(8x3x3)dx=\int_0^2 x^4(2x-x)+\cfrac13(8x^3-x^3) dx =


02x5+73x3dx=[16x6+712x4]02=\int_0^2 x^5+\cfrac73x^3 dx = \big[ \cfrac16x^6 + \cfrac7{12}x^4 \big]_0^2 =


16(2606)+712(2404)=22(23+7)3=20\cfrac16(2^6 - 0^6) + \cfrac7{12}(2^4 - 0^4) = \cfrac{2^2(2^3 + 7)}3 = 20



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