Mathematical Tools for Physics - Department of Physics - University

8—Multivariable Calculus 190

θis not in a plane passing through the origin. It is in a plane parallel to thex-yplane, so it has a

radiusrsinθ.

rectangular cylindrical spherical

volume d^3 r= dxdydz rdrdφdz r^2 sinθdrdθdφ

area d^2 r= dxdy rdφdz or rdφdr r^2 sinθdθdφ

Examples of Multiple Integrals
Even in rectangular coordinates integration can be tricky. That’s because you have to pay attention to
the limits of integration far more closely than you do for simple one dimensional integrals. I’ll illustrate
this with two dimensional rectangular coordinates first, and will choose a problem that is easy but still
shows what you have to look for.

An Area

Find the area in thex-yplane between the curvesy=x^2 /aandy=x.

(A)

∫a

0

dx

∫x

x^2 /a

dy 1 and (B)

∫a

0

dy

∫√ay

y

dx 1

y

x

y

x

In the first instance I fixxand add the pieces ofdyin the strip indicated. The lower limit of the

dyintegral comes from the specified equation of the lower curve. The upper limit is the value ofyfor

the givenxat the upper curve. After that the limits on the sum overdxcomes from the intersection

of the two curves:y=x=x^2 /agivesx=afor that limit.

In the second instance I fixyand sum overdxfirst. The left limit is easy,x=y, and the upper

limit comes from solvingy=x^2 /aforxin terms ofy. When that integral is done, the remainingdy

integral starts at zero and goes up to the intersection aty=x=a.

Now do the integrals.

(A)

∫a

0

dx

[

x−x^2 /a

]

=

a^2

2

−

a^3

3 a

=

a^2

6

(B)

∫a

0

dy

[√

ay−y

]

=a^1 /^2

a^3 /^2

3 / 2

−

a^2

2

=

a^2

6

If you would care to try starting this calculation from the beginning,withoutdrawing any pictures, be
my guest.