Consider the double integral I = R x^2y-x dA where R is the first quadrant regio
ID: 2898180 • Letter: C
Question
Consider the double integral I =R x^2y-x dA where R is the first quadrant region enclosed by the curves y = 0, y = x^2, y = 2-x.
(a) Sketch the region of integration.
(b) Express the integral I as an iterated integral.
Explanation / Answer
Consider the double integral I =??R x^2y-x dA where R is the fi rst quadrant region enclosed by the curves y = 0, y = x^2, y = 2-x. (a) Sketch the region of integration. x+y=2==> y=2-x y=x^2 y=y==> x^2=2-x ==> x^2+x-2=0 ==> (x+2)(x-1)=0 x=-2 or x=1 There are two regions R1) y=x^2, y=0 and x=1 R2) y=2-x, y=0 and x=1 (b) Express the integral I as an iterated integral. =??R x^2y-x dA = int_{x=0}^{1} int_{y=0}^{x^2} (x^2y-x)dydx + int_{x=1}^{2} int_{y=0}^{2-x} (x^2y-x)dydx = (-5/28) + (-2/5) = -81/140