Natural Science Department Duy Tan University Double integrals

Natural Science Department – Duy Tan University Double integrals over Type I Regions In this section, we will learn: How to evaluate double integrals over type 1 regions. Lecturer: Ho Xuan Binh Da Nang-12/2014

Natural Science Department – Duy Tan University 1 TYPE I REGION A plane region D is said to be of type 1 if it lies between the graphs of two continuous functions of x, that is, D = {(x, y) | a ≤ x ≤ b, g 1(x) ≤ y ≤ g 2(x)} where g 1 and g 2 are continuous on [a, b]. Double integrals over type 1 regions

Natural Science Department – Duy Tan University 2 TYPE I REGIONS Double integrals over type 1 regions

Natural Science Department – Duy Tan University 3 Double integrals over Type I Regions If f is continuous on a type 1 region D such that D = {(x, y) | a ≤ x ≤ b, g 1(x) ≤ y ≤ g 2(x)} then Double integrals over type 1 regions

Natural Science Department – Duy Tan University 4 Example v. Evaluate vwhere D is the region bounded by the parabolas y = 2 x 2 and y = 1 + x 2. Double integrals over type 1 regions

Natural Science Department – Duy Tan University 5 Hint D = {(x, y) | – 1 ≤ x ≤ 1, 2 x 2 ≤ y ≤ 1 + x 2} Double integrals in Polar Coordinates

Natural Science Department – Duy Tan University 6 Solution Double integrals over type 1 regions

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