Let c,r be constants, and D={(x,y,z):x^2+y^2+z^2≤r^2}. The answer to ∭D cdV
is
Select one:
a. (π^2cr^3)/3
b. (4πcr^3)/3
c. 4πcr^3
d. (πcr^4)/3
e. (4πcr^2)/2
f. (4r^3)/3
g. (πcr^3)/3
1
Expert's answer
2020-06-07T16:19:00-0400
The easiest way to do this is to make a switch to spherical coordinates. There ρ2=x2+y2+z2 anddxdydz=ρ2sinϕdρdθdφ.∭Dcdv=∭Dcdxdydz=∭Dc⋅ρ2sinφdρdθdφ Now we are integrating over a regionD.What isD? It is a sphere of radius r centered at the origin. So 0≤ρ≤r,0≤θ≤2π , and 0≤φ≤π∭Dcρ2sinφdρdθdφ=∫0π∫02π∫0rcρ2sinφdρdθdφc∫02πdθ∫0πsinφdφ∫0rρ2dρ=c(θ∣∣02π)(−cosφ∣∣0π)(3ρ3∣∣0r)=c(2π)(−(−1−1))(3r3)=34πcr3The answer is b
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