Transcription of Practice problems on double integrals
1 Math 461 Introduction to HildebrandPractice problems on double integralsThe problems below illustrate the kind of double integrals that frequently arise in probability first group of questions asks to set up a double integral of a general functionf(x,y) over a giving regionin thexy-plane. This means writing the integral as an iterated integral of the form f(x,y)dxdyand/or f(x,y)dydx, with specific limits in place of the asterisks. To do this, follow the steps above (mostimportantly, sketch the given region). The remaining questions are evaluations of integrals over Set up a double integral off(x,y) over the region given by 0<x<1,x<y<x+ : 1x=0 x+1y=xf(x,y)dydx2. Set up a double integral off(x,y) over the part of the unit square 0 x 1,0 y 1, on whichy : 1x=0 x/2y=0f(x,y)dydxor 1/2y=0 1x=2yf(x,y)dxdy3. Set up a double integral off(x,y) over the part of the unit square on whichx+y> : 1/2x=0 1y=1/2 xf(x,y)dydx+ 1x=1/2 1y=0f(x,y)dydx4. Set up a double integral off(x,y) over the part of the unit square on whichbothxandyare greaterthan : 1x=1/2 1y=1/2f(x,y)dydx5.
2 Set up a double integral off(x,y) over the part of the unit square on whichat least one ofxandyis greater than : 1/2x=0 1y=1/2f(x,y)dydx+ 1x=1/2 1y=0f(x,y)dydx6. Set up a double integral off(x,y) over the part of the region given by 0<x<50 y<50 on whichbothxandyare greater than : 30x=20 50 xy=20f(x,y)dydx7. Set up a double integral off(x,y) over the set of all points (x,y) in the first quadrant with|x y| : 1x=0 x+1y=0f(x,y)dydx+ x=1 x+1y=x 1f(x,y)dydx1 Math 461 Introduction to Hildebrand8. Evaluate Re x ydxdy, whereRis the region in the first quadrant in whichx+y : 1x=0 1 xy=0e xe ydydx= 10e x(1 e (1 x))dx= 10(e x e 1)dx= 1 2e Evaluate Re x 2ydxdy, whereRis the region in the first quadrant in whichx ySolution: x=0 y=xe x 2ydydx= 012e 3xdx=1610. Evaluate R(x2+y2)dxdy, whereRis the region 0 x y LSolution: Ly=0 yx=0(x2+y2)dydx= Ly=0(13x3+y2x) yx=0dydx= L043y3dy= Evaluate R(x y+ 1)dxdy, whereRis the region inside the unit square in whichx+y : 1y= x(x y+ 1)dydx+ 1x= 1y=0(x y+ 1)dydx= (xy 12y2+y) 1y= xdx+ 1x= (xy 12y2+y) 1y=0dx= (x(1 12+x) 12(1 (12 x)2) + (1 12+x))dx+ (x+12)dx= (18+x+32x2)dx+(12x2+12x) 18+12 22+13 23 32+38+14=7812.
3 Evaluate 10 10xmax(x,y) : 1x=0 xy=0x2dydx+ 1x=0 1y=xxydydx= 10(x3+x1 x22)dx=14+12(12 14)=
