Use the result of Exercise 31 to find the volume of each domeshapedsolid lying below the

Chapter 14, Problem 32

(choose chapter or problem)

Use the result of Exercise 31 to find the volume of each dome shaped solid lying below the surface z = f(x, y) and above the elliptical region R. (Hint: After making the change of variables given by the results in Exercise 31, make a second change of variables to polar coordinates.)

(a) \(f(x,\ y)=16-x^2-y^2\)

\(R:\ \frac{x^2}{16}+\frac{y^2}{9}\ \leq\ 1\)

(b) \(f(x,\ y)=A \cos (\frac{\pi}{2}\sqrt{\frac{x^2}{a^2}+\frac{y^2}{b^2}})\)

\(R:\ \frac{x^2}{a^2}+\frac{y^2}{b^2}\ \leq\ 1\)

Text Transcription:

f(x, y)=16-x^2 -y^2

R:x^2 /16 +y^2 /9 leq 1

f(x, y)=A cos (pi/2 sqrtx^2/a^2 +y^2/b^2 )

R: x^2 /a^2 +y^2 /b^2 leq 1