Calc 3 Lecture Notes
Section 13.1
Page 1 of 9
Section 13.1: Double Integrals
Big idea: To compute the volume of a solid bounded by a surface z = f(x, y) and a region in the
x-y plane, we can integrate in one direction to find the cross-sectional area of thin slices of the
solid, then integrate in the other direction to find the volume of the solid.
Big skill: You should be able to compute the double integral of a function of two variables for
various bounded regions in the x-y plane.
Definition 1.1: The Definite Integral of a Function of a Single Variable
For any function f defined on the interval [a, b] and ||P|| (the norm of the partition) defined as the
maximum of all the intervals on [a, b] (i.e., ||P|| = max{xi}), the definite integral of f on [a, b]
is:
b

f  x  dx  lim
|| P|| 0
a
n
 f  c  x ,
i 1
i
i
provided the limit exists and is the same for all values of the evaluation points ci  [xi-1, xi] for
i = 1, 2, …, n. In this case, we say f is integrable on [a, b].
Riemann Sum Over a Region:
Vi  Height  Base Area  f  ui , vi  Ai
n
n
i 1
i 1
V   Vi   f  ui , vi  Ai
Calc 3 Lecture Notes
Practice:
Section 13.1
Page 2 of 9
 y 
1. Approximate the volume of the solid bounded by the surface z  x 2 sin 
 , the
 6 
rectangle R = {(x, y) | 0 ≤ x ≤ 6, 0 ≤ y ≤ 6} in the x-y plane, and the plane x = 6. Use four
square partitions.
Calc 3 Lecture Notes
Section 13.1
Page 3 of 9
Definition 1.2: Double Integral of a Function of Two Variables Over a Rectangular Region
For any function f(x, y) defined on the rectangle R = {(x, y) | a ≤ x ≤ b, c ≤ y ≤ d}, and ||P|| (the
norm of the partition) defined as the maximum diagonal of any rectangle in the partition, the
double integral of f over R is defined as:

R
f  x, y  dA  lim
|| P||0
n
 f u , v  A ,
i 1
i
i
i
provided the limit exists and is the same for all choices of the evaluation points (ui, vi)  R for
i = 1, 2, …, n. In this case, we say f is integrable over R.
To compute a double integral, we can think of taking thin slices of the solid parallel to the y-z
plane and integrating the function while holding x constant to find A(x) (i.e., perform a partial
integration), and then integrating the volume of all the thin slices to find the total volume. This
is called an iterated integral. Likewise, we could take thin slices parallel to the x-z plane to find
A(y), and then integrate the volume of those slices to find total volume. In either case, we get the
same answer
Calc 3 Lecture Notes
Section 13.1
Page 4 of 9
Practice:
 y 
2. Compute the exact volume of the solid bounded by the surface z  x 2 sin 
 , the
 6 
rectangle R = {(x, y) | 0 ≤ x ≤ 6, 0 ≤ y ≤ 6} in the x-y plane, and the plane x = 6, first using
thin slices parallel to the y-z plane, then using thin slices parallel to the x-z plane. Sketch
some sample thin slices on the diagrams below.
Theorem 1.1: Fubini’s Theorem (Order of Integration is Interchangeable)
If a function f(x, y) is integrable on the rectangle R = {(x, y) | a ≤ x ≤ b, c ≤ y ≤ d}, then we can
write the double integral of f over R as either of the iterated integrals:
b d
d b
a c
c a
 f  x, y  dA    f  x, y  dydx    f  x, y  dxdy .
R
Calc 3 Lecture Notes
Section 13.1
Page 5 of 9
3. Compute the exact volume of the solid bounded by the surface z  x 2  y 2 above the
region R = {(x, y) | -1 ≤ x ≤ 1, -1 ≤ y ≤ 1} in the x-y plane. Use symmetry if possible to
simplify the integral.
2 1
4. Sketch the solid whose volume is given by the iterated integral
   2  x  2 y  dydx
0 1
Calc 3 Lecture Notes
Section 13.1
Page 6 of 9
Now: What if the region in the x-y plane is not a rectangle?
Definition 1.3: Double Integral of a Function of Two Variables Over Any Bounded Region
For any function f(x, y) defined on a bounded region R  2 , we define the double integral of f
over R as:

R
f  x, y  dA  lim
|| P||0
n
 f u , v  A ,
i 1
i
i
i
provided the limit exists and is the same for all choices of the evaluation points (ui, vi)  Ri for
i = 1, 2, …, n. In this case, we say f is integrable over R.
Theorem 1.2: Double Integral Over a Region with Nonconstant Bounds in the x Direction
If a function f(x, y) is continuous on a bounded region R defined by
R = {(x, y) | a ≤ x ≤ b, g1(x) ≤ y ≤ g2(x)} for continuous functions g1 and g2 where g1(x) ≤ g2(x)
for all x  [a, b], then:
b g2  x 
 f  x, y  dA    f  x, y  dydx .
R
a g1  x 
Calc 3 Lecture Notes
Section 13.1
Page 7 of 9
Practice:
5. Compute the double integral of the function f  x, y   x2  y 2 over the region bounded
by the curves y = 1 – x2 and y = x2 - 1 in the x-y plane.
Calc 3 Lecture Notes
Section 13.1
Page 8 of 9
Theorem 1.3: Double Integral Over a Region with Nonconstant Bounds in the y Direction
If a function f(x, y) is continuous on a bounded region R defined by
R = {(x, y) | h1(y) ≤ x ≤ h2(y)} c ≤ y ≤ d} for continuous functions h1 and h2 where h1(x) ≤ h2(x)
for all y  [c, d], then:
d h2  y 
 f  x, y  dA    f  x, y  dxdy .
R
c h1  y 
Practice:
6. Compute the double integral of the function f  x, y   6  x  2 y over the region
bounded by the curves x = y2 and y = 2 – x in the x-y plane.
Calc 3 Lecture Notes
Section 13.1
Page 9 of 9
1 1
7. Evaluate the iterated integral
e
x2
dxdy by switching the order of integration.
0 y
Theorem 1.4: Linear Combinations of Double Integrals
Let the function f(x, y) and g(x, y) be integrable over the region R R 
constant. Then the following hold:
i.
 cf  x, y  dA  c f  x, y  dA
R
ii.
R
  f  x, y   g  x, y  dA   f  x, y  dA   g  x, y  dA
R
iii.
2
R
R
if R = R1  R2, where R1 and R2 are nonoverlapping regions, then
 f  x, y  dA   f  x, y  dA   f  x, y  dA .
R
R1
R2
, and let c be any