Only question 8 & 9 plz. They are related. 7. Evaluate using cylindrical coordinates:[[[ z dxdydz; T:0sxs1,0sysvi-x,0szs1-x -yа.6.b.16с.3d.2е.12f. None of these8. Write the formula for the volume using cylindrical coordinates for the solid that is boundedabove by x' +y² +z? = 25 and below by z = Vx + y +125-r"I r dzdrdeа.b. "I r dzdrder+1V25-r dzdrdeс.r+ld.2 4 cv25-r dzdrder+l25-rCI r dzdrdof. None of theseе.r+l9. Find the volume for the solid given in number 8.41а.340b.43с.341d.4e. none of these3.

The solid is bounded by two surfaces: S: x2+y2+z2=25C: z = x2+y2+1 Plotted on graph, it looks like…

Answered: 8. Write the formula for the volume…

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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Only question 8 & 9 plz. They are related.

7. Evaluate using cylindrical coordinates:
[[[ z dxdydz; T:0sxs1,0sysvi-x,0szs1-x -y
а.
6.
b.
16
с.
3
d.
2
е.
12
f. None of these
8. Write the formula for the volume using cylindrical coordinates for the solid that is bounded
above by x' +y² +z? = 25 and below by z = Vx + y +1
25-r
"I r dzdrde
а.
b. "I r dzdrde
r+1
V25-
r dzdrde
с.
r+l
d.
2 4 cv25-
r dzdrde
r+l
25-r
CI r dzdrdo
f. None of these
е.
r+l
9. Find the volume for the solid given in number 8.
41
а.
3
40
b.
43
с.
3
41
d.
4
e. none of these
3.

Expert Solution

Step 1

The solid is bounded by two surfaces:
$S : x^{2} + y^{2} + z^{2} = 25 C : z = \sqrt{x^{2} + y^{2}} + 1$

Plotted on graph, it looks like

Step 2

First we try and determine the curve of intersection. We substitute the equation of sphere into the equation of cone, and get,
$z^{2} = x^{2} + y^{2} + 1 + 2 \sqrt{x^{2} + y^{2}} \Rightarrow z^{2} = 25 - z^{2} + 1 + 2 \sqrt{25 - z^{2}} \Rightarrow z^{2} - 13 = \sqrt{25 - z^{2}} \Rightarrow z^{4} + 169 - 26 z^{2} = 25 - z^{2} \Rightarrow z^{4} + 144 - 25 z^{2} = 0 \Rightarrow z^{4} - 9 z^{2} - 16 z^{2} + 144 = 0 \Rightarrow z^{2} (z^{2} - 9) - 16 (z^{2} - 9) = 0 \Rightarrow (z^{2} - 9) (z^{2} - 16) = 0 \Rightarrow z^{2} = 9; z^{2} = 16 \Rightarrow z = \pm 3, \pm 4$
Since the cone is formed on the positive axis, can eliminate z= -3, and z = -4.

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