Questions d, e 2. Consider the triple integralV9-x²18-т?—у21 dz dy dx/9-x2x²+y²(a) Sketch the region of integration.(b) Explain the significance of the integrand for this integral and its region.(c) Rewrite the integral using cylindrical coordinates.(d) Rewrite the integral using spherical coordinates.(e) Find the exact value of either of the integrals.

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Description: Questions d, e 2. Consider the triple integralV9-x²18-т?—у21 dz dy dx/9-x2x²+y²(a) Sketch the region of integration.(b) Explain the significance of the integrand for this integral and its region.(c) Rewrite the integral using cylindrical coordinates.

d) First, we write down the given limits. -3≤x≤3-9-x2≤y≤9-x2x2+y2≤z≤18-x2-y2The first two…

Answered: 2. Consider the triple integral c3…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Questions d, e

2. Consider the triple integral
V9-x²
18-т?—у2
1 dz dy dx
/9-x2
x²+y²
(a) Sketch the region of integration.
(b) Explain the significance of the integrand for this integral and its region.
(c) Rewrite the integral using cylindrical coordinates.
(d) Rewrite the integral using spherical coordinates.
(e) Find the exact value of either of the integrals.

Expert Solution

Step 1

d) First, we write down the given limits.
$- 3 \leq x \leq 3 - \sqrt{9 - x^{2}} \leq y \leq \sqrt{9 - x^{2}} \sqrt{x^{2} + y^{2}} \leq z \leq \sqrt{18 - x^{2} - y^{2}}$

The first two inequalities tells us that the region on X-Y plane is a circular disk, of radius 3.
So, we have $θ$ (angle around z-axis) as $0 \leq θ \leq 2 π$ .

Now, the limits of z. The lower limit $z = \sqrt{x^{2} + y^{2}}$ is the upper half of the cone, and the upper bound $z = \sqrt{18 - x^{2} - y^{2}}$ is the upper part of the sphere $x^{2} + y^{2} + z^{2} = 18$ , with radius $3 \sqrt{2} .$

So, the second spherical co-ordinate $ρ$ has the range $0 \leq ρ \leq 3 \sqrt{2}$ .

Step 2

To find the range of the third spherical co-ordinate, we find where the sphere and the cone intersect.

$x^{2} + y^{2} + z^{2} = 18 \Rightarrow z^{2} + z^{2} = 18 \Rightarrow z^{2} = 9 \Rightarrow z = 3$

This gives
$ρ \cos φ = z \Rightarrow 3 \sqrt{2} \cos φ = 3 \Rightarrow \cos φ = \frac{1}{\sqrt{2}} \Rightarrow φ = \frac{π}{4}$
So, the range is $0 \leq φ \leq \frac{π}{4}$ .

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