Express the following rectangular triple integral as a triple integral in cylind

Question

Express the following rectangular triple integral as a triple integral in cylindrical coordinates. integral_-8^8 integral-0^Squareroot 64-x^2 integral_0^64-x^2 - y^2 2Squarerootx^2 + y^2 dz dx = integral_theta1^theta2 integral_r1^r2 integral_z1^z2 dz dr dtheta z_1 = z_2 = r_1 = r_2 = theta_1 = theta_2 = Evaluate the integral integral_-8^8 integral_0^squareroot 64-x^2 integral_0^64-x^2 - y^2 2Squarerootx^2 + y^2 dz dy dx = If you don't get this in 3 tries, you can see a similar example (online). However, try to use this as a last resort or after you have already solved the problem. There are no "see a similar examples" on the Exams!

Explanation / Answer

0<=z<=64-x2-y2, 0<=y<=(64-x2) ,-8<=x<=8

in cylindrical coordinates

x=rcos, y=rsin

x2+y2=r2

0=64-x2-y2

0=64-r2

r=8

0<=<=,0<=r<=8 ,0<=z<=64-r2

dz dy dx=r dz dr d

1=0

2=

r1=0

r2=8

z1=0

z2=64-r2

[0 to ] [0 to 8] [0 to 64-r2] 2r2 r dz dr d

[0 to ] [0 to 8] [0 to 64-r2] 2r2 dz dr d

[0 to ] [0 to 8][0 to 64-r2] 2r2 z dr d

[0 to ] [0 to 8]2r2(64-r2-0) dr d

[0 to ] [0 to 8]2(64r2-r4) dr d

[0 to ][0 to 8]2(64(1/3)r3-(1/5)r5) d

[0 to ]2(64(1/3)83-(1/5)85)-0 d

[0 to ](131072/15) d

=[0 to ](131072/15)

=(131072/15)(-0)

=(131072/15)

=27451.7

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