Triple integrals in cylindrical and spherical coordinates
Triple integrals in cylindrical and spherical coordinates
Triple integrals in cylindrical and spherical coordinates
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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, V<br />
Theorem (change to <strong>spherical</strong> coord<strong>in</strong>ates <strong>in</strong> a tiple <strong>in</strong>tegral)<br />
Let<br />
E := {(ρ, θ, φ) : α ≤ θ ≤ β,<br />
Then:<br />
���<br />
E<br />
f (x, y, z) dV<br />
c ≤ φ ≤ d, ≤ g1(θ, φ) ≤ ρ ≤ g2(θ, φ)}.<br />
� d � β � g2(θ,φ)<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 />
g1(θ,φ)<br />
ρ 2 s<strong>in</strong> φ dρ dθ dφ.