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Using geometry, evaluate the double integral. doubleintegral_D sqrt(64 - x^2 - y^2) dA over the...

Question:

Using geometry, evaluate the double integral.

{eq}\iint_{D} \sqrt{64 - x^{2} - y^{2}}dA \text{ over the circular disk } D: x^{2} + y^{2} \leq 64 {/eq}

Double Integrals:

In much the same way integrating the difference between two curves gives us the area between them, integrating the difference between two surfaces gives us the volume between them. When the surface is one familiar from geometry, we can use geometry to evaluate the integral.

Answer and Explanation:

Let' think about what the integral is describing. The top surface is

{eq}\begin{align*} z &= \sqrt(64 - x^2 - y^2) \\ z^2 &= 64 - x^2 - y^2 \\ x^2...

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Double Integrals: Applications & Examples

from AP Calculus AB & BC: Help and Review

Chapter 12 / Lesson 14
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