Calc 3 Lecture NotesSection 13.1Page 1 of 10

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:

,

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 fis integrable on [a, b].

Riemann Sum Over a Region:

Practice:

  1. Approximate the volume of the solid bounded by the surface , the rectangle R = {(x, y) | 0 ≤ x ≤ 6, 0 ≤ y ≤ 6} in the x-y plane, and the plane x = 6. Use four square partitions.

Definition 1.2: Double Integral of a Function of Two Variables Over a Rectangular Region

For any function f(x, y) defined on the rectangleR = {(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 foverRis defined as:

,

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 integrableover 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

Practice:

  1. Compute the exact volume of the solid bounded by the surface , the 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:

.

  1. Compute the exact volume of the solid bounded by the surface 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.
  1. Sketch the solid whose volume is given by the iterated integral

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, we define the double integral off over R as:

,

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:

.

Practice:

  1. Compute the double integral of the function over the region bounded by the curves y = 1 – x2 and y = x2 - 1 in the x-y plane.

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:

.

Practice:

  1. Compute the double integral of the function over the region bounded by the curves x = y2 and y = 2 – x in the x-y plane.

  1. Evaluate the iterated integral by switching the order of integration.

Theorem 1.4: Linear Combinations of Double Integrals

Let the function f(x, y) and g(x, y) be integrable over the region RR, and let c be any constant. Then the following hold:

  1. if R = R1R2, where R1 and R2 are nonoverlapping regions, then .