Triple integrals in cylindrical and spherical coordinates
Triple integrals in cylindrical and spherical coordinates
Triple integrals in cylindrical and spherical coordinates
Create successful ePaper yourself
Turn your PDF publications into a flip-book with our unique Google optimized e-Paper software.
MATH 209—<br />
Calculus,<br />
III<br />
Volker Runde<br />
<strong>Triple</strong><br />
<strong><strong>in</strong>tegrals</strong> <strong>in</strong><br />
cyl<strong>in</strong>drical<br />
coord<strong>in</strong>ates<br />
<strong>Triple</strong><br />
<strong><strong>in</strong>tegrals</strong> <strong>in</strong><br />
<strong>spherical</strong><br />
coord<strong>in</strong>ates<br />
Integration <strong>in</strong> <strong>spherical</strong> coord<strong>in</strong>ates, IV<br />
Consequence<br />
���<br />
E<br />
f (x, y, z) dV = lim<br />
n,m,ν→∞<br />
n�<br />
m�<br />
ν�<br />
j=1 k=1 ℓ=1<br />
f (ρ ∗ j s<strong>in</strong> φ ∗ ℓ cos θ∗ k , ρ∗ j s<strong>in</strong> φ ∗ ℓ s<strong>in</strong> θ∗ k , ρ∗ j cos φ ∗ ℓ )ρ2 s<strong>in</strong> φ∆ρ∆θ∆φ<br />
� d � β � b<br />
=<br />
f (ρ s<strong>in</strong> φ cos θ, ρ s<strong>in</strong> φ s<strong>in</strong> θ, ρ cos φ)<br />
c<br />
α<br />
a<br />
ρ 2 s<strong>in</strong> φ dρ dθ dφ.