Explanation Given : ∫ ∫ D x d A , the figure given below showing the region D. Calculation : Step 1: We compute the integral using decomposition of a domain into smaller domains. We denote by D 2 and D 1 , the right semicircles of radius 2 and 1 respectively. Step 2: Then D 2 is the union of D 1 and D , which do not overlap except on a boundary curve. Therefore, ∫ ∫ D 2 x d A = ∫ ∫ D 1 x d A + ∫ ∫ D x d A ∫ ∫ D x d A = ∫ ∫ D 2 x d A − ∫ ∫ D 1 x d A Step 3: To express D 1 and D 2 as horizontally simple regions, we write the equations of the circles in the form x = 1 − y 2 and x = 4 − y 2 , respectively. The equations describing the regions D 1 and D 2 are − 2 ≤ y ≤ 2 , -1 ≤ y ≤ 1 0 ≤ x ≤ 4 − y 2 0 ≤ x ≤ 1 − y 2 D 2 D 1

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Question

Chapter 16.2, Problem 44E

To determine

The value of $\int \int_{D} x d A$ for the region D in the given figure

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Chapter 16.2, Problem 43E

Chapter 16.2, Problem 45E

Chapter 16 Solutions

CALCULUS W/SAPLING ACCESS >IC<

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