sheet26.pdf

course: CalculusII

May14,2007

Problemsheet:Multipleintegrals

1.Evaluatethefollowingdoubleintegrals:

(a)

xy 2 dxdywhereistheregionboundedbyy 2 =2xandx=2;

(b)

x 2 +y 2 dxdywhereDisaregionboundedbyy=x,y=x+2,y=2,y=6

(c)

|x|+|y|dxdxdywhereD={(x,y):|x|+|y|1};

(d)

x+ydxdywhereregionDisboundedbyy 2 =2x,x+y=4,x+y=12;

(e)

xydxdywhereregionDisboundedbyxy=1x+y=5/2

(f)

cos(x+y)dxdywhereD={(x,y):0x;0y}

p |y−x 2 |dxdywhereD={(x,y):|x|1,0y2}

(g)

2.Evaluateintegralsusingpolarcoordinatetransformation:

(a)

p x 2 +y 2 dxdywhereD={(x,y):x 2 +y 2 2};

sin( p x 2 +y 2 )dxdywhereD={(x,y): 2 p x 2 +y 2 4 2 }

(b)

(c)

4 − y 2

9 dxdywhereD={(x,y): x 2

4 + y 2

9 =1}

3.Findvolumeoftheboundedbythesurfaces:

(a)z=1+x+y,z=0,x+y=1,x=0,y=0;

(b)x=y+z=5,x 2 +y 2 =1,x=0,y=0,z=0;

(c)z=x 2 +y 2 ,y=x 2 ,y=1,z=0;

(d)z 2 =xy,x 2 +y 2 =1;

(e)z=e − (x 2 +y 2 ,z=0,x 2 +y 2 =1;

(f)z=x 2 +y 2 ,z=x+y;

1− x 2