CHAPTER 13 Multiple Integrals Slide 2 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 13.1DOUBLE INTEGRALS 13.2AREA, VOLUME AND CENTER OF MASS 13.3DOUBLE INTEGRALS IN POLAR COORDINATES 13.4SURFACE AREA 13.5TRIPLE INTEGRALS 13.6CYLINDRICAL COORDINATES 13.7SPHERICAL COORDINATES 13.8CHANGE OF VARIABLES IN MULTIPLE INTEGRALS

13.4SURFACE AREA Preliminaries Slide 3 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

13.4SURFACE AREA Preliminaries Slide 4 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Let the dimensions of R i be x i and y i, and let the vectors a i and b i form two adjacent sides of the parallelogram T i. Recall from our discussion of tangent planes in section 13.4 that the tangent plane is given by

13.4SURFACE AREA Preliminaries Slide 5 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

13.4SURFACE AREA Preliminaries Slide 6 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

13.4SURFACE AREA Preliminaries Slide 7 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

EXAMPLE 13.4SURFACE AREA 4.1Calculating 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 = y 2 + 4x lying above the triangular region R in the xy- plane with vertices at (0, 0), (0, 2) and (2, 2).

EXAMPLE Solution 13.4SURFACE AREA 4.1Calculating Surface Area Slide 9 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

EXAMPLE Solution 13.4SURFACE AREA 4.1Calculating Surface Area Slide 10 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

EXAMPLE 13.4SURFACE AREA 4.2Finding 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 + x 2 + y 2 that lies below the plane z = 5.

EXAMPLE Solution 13.4SURFACE AREA 4.2Finding Surface Area Using Polar Coordinates Slide 12 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

EXAMPLE Solution 13.4SURFACE AREA 4.2Finding Surface Area Using Polar Coordinates Slide 13 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

EXAMPLE 13.4SURFACE AREA 4.3Surface 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 − x 2 − y 2 that lies above the triangular region R in the xy- plane with vertices at the points (0, 0), (1, 1) and (1, 0).

EXAMPLE Solution 13.4SURFACE AREA 4.3Surface Area That Must Be Approximated Numerically Slide 15 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

EXAMPLE Solution 13.4SURFACE AREA 4.3Surface Area That Must Be Approximated Numerically Slide 16 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.