Chapter 15 Multiple Integrals 15.1 Double Integrals over Rectangles 15.2 Iterated Integrals 15.3 Double Integrals over General Regions 15.4 Double Integrals in polar coordinates 15.5* Applications of Double Integrals 15.6* Surface Area

15.1 Double Integrals over Rectangles Volumes and Double Integrals A function f of two variables defind on a closed rectangle and we suppose that The graph of f is a surface with equation Let S be the solid that lies above R and under the graph of f ,that is ,

(See Figure 1)Find the volume of S 1) Partition: The first step is to divide the rectangle R into subrectangles. Each with area

2) Approximation: A thin rectangular box: Base: Hight: We can approximate by

3) Sum: A double Riemann sum 4) Limit:

The sufficient condition of integrability: Definition The double integral of f over the rectangle R is if this limit exists. The sufficient condition of integrability: Theorem1. is integral on R

Theorem2. and f is discontinuous only on a finite number of smooth curves is integral on

Example 1 If evaluate the integral Solution

15.2 Iterated Integrals Partial integration with respect to y defines a function of x: We integrate A with respect to x from x=a to x=b, we get

The integral on the right side is called an iterated integral and is denoted by Thus Similarly

Fubini’s theorem If f is continuous on the rectangle then More generally, this is true that we assume that f is bounded on R , f is discontinuous only on a finite number of smooth curves, and the iterated integrals exist. The proof of Fubini’s theorem is too difficult to include In our class.

If f (x,y) ≥0,then we can interpret the double integral as the volume V of the solid S that lies above R and under the surface z=f(x, y). So

Or

Example 1　　　　　　　 Solution 1

Solution 2

Example 2　　　　　　　 Solution

Example 3　　　　　　　 Solution

Specially If Then

15.3 Double Integrals over General Regions Suppose that D is a bounded region which can be enclosed in a rectangular region R. D D A new function F with domain R:

If the integral of F exists over R, then we define the double integral of f over D by

If the integral of F exists over R, then we define the double integral of f over D by A plane region D is said to be of type I if Where and are continuous on [a,b] Some examples of type I

Evaluate where D is a region of type I A new function F with domain R:

If f is continuous on type I region D such that then

A plane region D is said to be of type II if Where and are continuous on [a,b] Some examples of type II

If f is continuous on type II region D such that then

Example 1 Evaluate ,where D is the region bounded by the parabolas Solution Type I

Type II

Properties of Double Integral Suppose that functions f and g are continuous on a bounded closed region D. Property 1 The double integral of the sum (or difference) of two functions exists and is equal to the sum (or difference) of their double integrals, that is, Property 2 Property 3 where D is divided into two regions D1 and D2 and the area of D1 ∩ D2 is 0.

Property 4 If f(x, y) ≥0 for every (x, y) ∈D, then Property 5 If f(x, y)≤g(x, y) for every (x, y) ∈D, then Moreover, since it follows from Property 5 that hence

Property 6 Suppose that M and m are respectively the maximum and minimum values of function f on D, then where S is the area of D. Property 7 (The Mean Value Theorem for Double Integral) If f(x, y) is continuous on D, then there exists at least a point (ξ,η) in D such that where S is the area of D. f (ξ,η) is called the average Value of f on D

Example 2 Evaluate ,where D is the region bounded by the parabolas Solution Type II

Type I

Example 3 Evaluate ,where D is the region bounded by the parabolas Solution Type I

15.4 Double Integrals in polar coordinates A polar rectangle

where The “center” of the polar subrectangle has polar coordinates

The area of is

Change of polar coordinate in a double integral If f is continuous on a polar rectangle R given by , ,where , then

1.If f is continuous on a polar region of the form then

2.If f is continuous on a polar region of the form then

3.If f is continuous on a polar region of the form then

Evaluate ,where D is the region bounded by the circles Example 1 Solution 1 2 o

Example 2 Evaluate where D = Solution In polar coordinates, the equation x2 + y2 –2x = 0 becomes The disc D is given by D’: Therefore