Calculus- Early Transcendentals, 2021a

494 Multiple Integration
Figure 14.6: Intersection of three cylinders.
Exercise 14.1.31 Prove that if f (x,y) is integrable and if g(x,y) =
f (x,y).
∫ x ∫ y
a b
Exercise 14.1.32 Reverse the order of integration on each of the following integrals
f (s,t) dt ds then g xy = g yx =
(a)
(b)
(c)
(d)
∫ 9 ∫ √ 9−y
0 0
∫ 2 ∫ lnx
1 0
∫ 1 ∫ π/2
0 arcsiny
∫ 1 ∫ 4
0 4x
f (x,y) dx dy
f (x,y) dy dx
f (x,y) dx dy
f (x,y) dy dx
(e)
∫ 3 ∫ √ 9−y 2
0 0
f (x,y) dx dy
Exercise 14.1.33 What are the parallels between Fubini’s Theorem and Clairaut’s Theorem?
14.2 Double Integrals in Polar Coordinates
Suppose we have a surface given in polar coordinates as z = f (r,θ) andwewishtofindtheintegralover
some region. We could attempt to translate into rectangular coordinates and do the integration there, but it
is often easier to stay in polar coordinates.
How might we approximate the volume under such a surface in a way that uses polar coordinates
directly? The basic idea is the same as before: we divide the region into many small regions, multiply
the area of each small region by the height of the surface somewhere in that little region, and add them
up. What changes is the shape of the small regions; in order to have a nice representation in terms of r