Mathematical Tools for Physics

8—Multivariable Calculus 222

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 ofdy in the strip indicated. The lower limit of thedy
integral comes from the specified equation of the lower curve. The upper limit is the value ofyfor the given
xat 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 remainingdyintegral 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.

b

a