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Page 1: CHAPTER 13 Multiple Integrals Slide 2 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 13.1DOUBLE INTEGRALS 13.2AREA,
Page 2: CHAPTER 13 Multiple Integrals Slide 2 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 13.1DOUBLE INTEGRALS 13.2AREA,

CHAPTER

13Multiple Integrals

13

Slide 2© The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

13.1 DOUBLE INTEGRALS13.2 AREA, VOLUME AND CENTER OF MASS13.3 DOUBLE INTEGRALS IN POLAR COORDINATES13.4 SURFACE AREA13.5 TRIPLE INTEGRALS13.6 CYLINDRICAL COORDINATES13.7 SPHERICAL COORDINATES13.8 CHANGE OF VARIABLES IN MULTIPLE INTEGRALS

Page 3: CHAPTER 13 Multiple Integrals Slide 2 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 13.1DOUBLE INTEGRALS 13.2AREA,

13.4 SURFACE AREA

Preliminaries

Slide 3© The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Page 4: CHAPTER 13 Multiple Integrals Slide 2 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 13.1DOUBLE INTEGRALS 13.2AREA,

13.4 SURFACE AREA

Preliminaries

Slide 4© The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Let the dimensions of Ri be xi and yi , and let the vectors ai and bi form two adjacent sides of the parallelogram Ti .

Recall from our discussion of tangent planes in section 13.4 that the tangent plane is given by

Page 5: CHAPTER 13 Multiple Integrals Slide 2 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 13.1DOUBLE INTEGRALS 13.2AREA,

13.4 SURFACE AREA

Preliminaries

Slide 5© The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Page 6: CHAPTER 13 Multiple Integrals Slide 2 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 13.1DOUBLE INTEGRALS 13.2AREA,

13.4 SURFACE AREA

Preliminaries

Slide 6© The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Page 7: CHAPTER 13 Multiple Integrals Slide 2 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 13.1DOUBLE INTEGRALS 13.2AREA,

13.4 SURFACE AREA

Preliminaries

Slide 7© The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Page 8: CHAPTER 13 Multiple Integrals Slide 2 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 13.1DOUBLE INTEGRALS 13.2AREA,

EXAMPLE

13.4 SURFACE AREA

4.1 Calculating Surface Area

Slide 8© The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Find the surface area of that portion of the surface z = y2 + 4x lying above the triangular region R in the xy-plane with vertices at (0, 0), (0, 2) and (2, 2).

Page 9: CHAPTER 13 Multiple Integrals Slide 2 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 13.1DOUBLE INTEGRALS 13.2AREA,

EXAMPLE

Solution

13.4 SURFACE AREA

4.1 Calculating Surface Area

Slide 9© The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Page 10: CHAPTER 13 Multiple Integrals Slide 2 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 13.1DOUBLE INTEGRALS 13.2AREA,

EXAMPLE

Solution

13.4 SURFACE AREA

4.1 Calculating Surface Area

Slide 10© The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Page 11: CHAPTER 13 Multiple Integrals Slide 2 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 13.1DOUBLE INTEGRALS 13.2AREA,

EXAMPLE

13.4 SURFACE AREA

4.2 Finding Surface Area Using Polar Coordinates

Slide 11© The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Find the surface area of that portion of the paraboloid z = 1 + x2 + y2 that lies below the plane z = 5.

Page 12: CHAPTER 13 Multiple Integrals Slide 2 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 13.1DOUBLE INTEGRALS 13.2AREA,

EXAMPLE

Solution

13.4 SURFACE AREA

4.2 Finding Surface Area Using Polar Coordinates

Slide 12© The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Page 13: CHAPTER 13 Multiple Integrals Slide 2 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 13.1DOUBLE INTEGRALS 13.2AREA,

EXAMPLE

Solution

13.4 SURFACE AREA

4.2 Finding Surface Area Using Polar Coordinates

Slide 13© The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Page 14: CHAPTER 13 Multiple Integrals Slide 2 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 13.1DOUBLE INTEGRALS 13.2AREA,

EXAMPLE

13.4 SURFACE AREA

4.3 Surface Area That Must Be Approximated Numerically

Slide 14© The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Find the surface area of that portion of the paraboloid z = 4 − x2 − y2 that lies above the triangular region R in the xy-plane with vertices at the points (0, 0), (1, 1) and (1, 0).

Page 15: CHAPTER 13 Multiple Integrals Slide 2 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 13.1DOUBLE INTEGRALS 13.2AREA,

EXAMPLE

Solution

13.4 SURFACE AREA

4.3 Surface Area That Must Be Approximated Numerically

Slide 15© The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Page 16: CHAPTER 13 Multiple Integrals Slide 2 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 13.1DOUBLE INTEGRALS 13.2AREA,

EXAMPLE

Solution

13.4 SURFACE AREA

4.3 Surface Area That Must Be Approximated Numerically

Slide 16© The McGraw-Hill Companies, Inc. Permission required for reproduction or display.


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