4) Set up integrals to find each of the following. Then evaluate the integrals.

Question

4) Set up integrals to find each of the following. Then evaluate the integrals. a) he average value of fox,y) 5x 2y 1 over the disc x2 y2 s5. (Use polar coordinates. b) The volume of the region between the paraboloids z x y -9 and z 9-x2 y (Use cylindrical coordinates.) c) The number of mosquitoes inside the sphere x y z' 9 (measured in feet) if the density of mosquitoes is d X2 y mosquitoes per cubic foot. (Use spherical coordinates.) d) The area inside the region between y x and y x. (Use rectangular coordinates.)

Explanation / Answer

d) We have given y=x^2 and y=x

Equating two curves x^2=x then (x^2-x)=0

x(x-1)=0 implies x=0 ,1

The area inside the region is

integration of (from x=0 to x=1){x^2-x}dx

{x3/3)-x2/2}(from x=0 to x=1)

(1/3)-(1/2)=-1/6

b) The two paraboloids intersect when x2+y2-9=9-x2-y2

2(x2+y2)-18=0 which implies x2+y2=9

then r=3 by Using cylindrical coordinates (r,theta,z) since r^2=x2+y2

Volume V=doubleintegration of {(x2+y2-9)-(9-x2-y2)}rdr(d theta) (from theta =0 to 2pi and r=0 to 3)

V=doubleintegration of {(r2-9)-(9-r2)}rdr(d theta) (from theta =0 to 2pi and r=0 to 3)

V=doubleintegration of {2r2-18}rdr(d theta) (from theta =0 to 2pi and r=0 to 3)

V=doubleintegration of {2r3-18r}dr(d theta) (from theta =0 to 2pi and r=0 to 3)

V=integration of {(r4/2)-9r2}(from  r=0 to 3) (d theta) (from theta =0 to 2pi)

V=integration of {(81/2)-81} (d theta) (from theta =0 to 2pi)

V=integration of {(-81/2)} (d theta) (from theta =0 to 2pi)={(-81/2)*theta}(from theta =0 to 2pi)

V=-81*(pi)

a) polar coordinates are x=rcos(theta) y=rsin(theta) f(x,y)=5x+2y+1 and disc x^2+y^2 lessthan or equalt to 5

r^2=5 and r=sqrt(5)

The average value of the function is

={1/(sqrt(5)-0)} * doubleintegration of (from theta =0 to 2pi and r=0 to sqrt(5)){5rcos(theta)+2(rsin(theta))+1}rdrd(theta)

={1/(sqrt(5)-0)}*doubleintegration of (from theta =0 to 2pi and r=0 to sqrt(5)){5*sqrt(5) cos(theta)+2*sqrt(5) sin(theta))+1}sqrt(5)drd(theta)

={1/(sqrt(5)-0)}*doubleintegration of (from theta =0 to 2pi and r=0 to sqrt(5)){25 cos(theta)+10sin(theta))+sqrt(5)}drd(theta)

={1/(sqrt(5)-0)}*integration of (from theta =0 to 2pi){25cos(theta)r+10sin(theta)r+sqrt(5)r}(from r=0 to sqrt(5) )d(theta)

={1/(sqrt(5)}*integration of (from  theta =0 to 2pi){sqrt(5)* 25cos(theta)+ sqrt(5) *10sin(theta)+5}d(theta)

={1/(sqrt(5)}*{sqrt(5)* 25sin(theta) - sqrt(5) *10 *cos(theta) +5(theta) }(from  theta =0 to 2pi)

={1/(sqrt(5)}*{(0- sqrt(5) *10+10pi)-(0- sqrt(5) *10)}={1/(sqrt(5)}*{10pi}

=(10*pi)/sqrt(5)=14.0496

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