Explanation Given : D is the domain bounded by y = x 2 + 1 and y = 2 Calculation : Step 1: We can write: m . A r e a ( D ) ≤ ∫ ∫ D f ( x , y ) d A ≤ M . A r e a ( D ) Where, m and M are constants such that m ≤ f ( x , y ) ≤ M Step 2: First let us compute the area of the domain D: A r e a = ∫ − 1 1 ∫ x 2 + 1 2 1 d y d x = ∫ − 1 1 2 − ( x 2 + 1 ) d y d x = ∫ − 1 1 1 − x 2 d y d x = x − 1 3 x 3 | − 1 1 = ( 1 − 1 3 ) − ( − 1 + 1 3 ) = 4 3 Step 3: Then we must bound f ( x , y ) = x 2 + y 2 by constants (above and below)...

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ISBN: 9781319050733

Author: Rogawski

Publisher: MAC HIGHER

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Question

Chapter 16.2, Problem 62E

To determine

The mathematical proof of the inequality $\frac{4}{3} \leq \int \int_{D} (x^{2} + y^{2}) d A \leq \frac{20}{3}$ where D is the domain bounded by $y = x^{2} + 1$ and $y = 2$

Expert Solution & Answer

Chapter 16 Solutions

CALCULUS (CLOTH)

Ch. 16.1 - Prob. 1PQ Ch. 16.1 - Prob. 2PQ Ch. 16.1 - Prob. 3PQ Ch. 16.1 - Prob. 4PQ Ch. 16.1 - Prob. 5PQ Ch. 16.1 - Prob. 6PQ Ch. 16.1 - Prob. 1E Ch. 16.1 - Prob. 2E Ch. 16.1 - Prob. 3E Ch. 16.1 - Prob. 4E

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