Chapter 15, section 1, Slides of Advanced Calculus

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Chapter 12Chapter 12 Dr. Phung Minh Duc National Economic University

MULTIPLE INTEGRALSMULTIPLE INTEGRALS

Contents 12.1. Double integrals over rectangles 12.2. Double Integrals over General Regions 12.3. Double Integrals in Polar Coordinates 12.4. Applications of Double Integrals 12.5. Triple Integrals 12.6. Triple Integrals in Cylindrical Coordinates 12.8. Change of Variables in Multiple Integrals

Double integrals over rectangles

Double integrals over rectangles

Double integrals over rectangles

Double integrals over rectangles

Example: Estimate the volume of the solid that lies above the square and below the elliptic paraboloid ଶ ଶ

. Divide into four equal squares and choose the sample point to be the upper right corner of each square (^) ௜௝. Sketch the solid and the approximating rectangular boxes.

Double integrals over rectangles

Iterated integrals

Suppose that is a function of two variables that is continuous on the rectangle We have

Double integrals over rectangles Example: Evaluate the iterated integrals

Double integrals over rectangles Example: Evaluate the iterated integrals

Fubini’s Theorem Example: Evaluate the double integrals ଶ ோ , where.

Properties of Double Integrals