Vx? + yZ 49x2 49 [F,(x, y)]² + [f,(x, y)]² 1+ x² + y2 e have [f,(x, y)]² + [f,(x, y)]² = /
Vx? + yZ 49x2 49 [F,(x, y)]² + [f,(x, y)]² 1+ x² + y2 e have [f,(x, y)]² + [f,(x, y)]² = /
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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![Tutorial Exercise
Find the area of the surface.
The portion of the cone z = 7
x² + y2 inside the cylinder x² + y2 = 9
Step 1
The definition of the surface area says if f and its first partial derivatives are continuous on the closed interval R in the xy-plane, then the area of the surface S given by z = f(x, y) over R is
S =
ds
= / | V1+ [f,(x, y)1² + [f,cx, v)]? ds.
We are asked to find the area of the portion of the cone z = 7VX2 + y² inside the cylinderx? + y2 = 9.
Step 2
To findf, (x, y), partially differentiate f(x, y) with respect to x.
x2
+ y
=
дх
7x
Vx² + y²
Similarly, find f,(x, y).
F,lx, y) = (7V;
x²
y2
ду
7
Therefore,
49x2
+
49
V1 + [f,(x, y)]² + [f,(x, y)]²
1 +
x2 + y2
x² + y2
Simplifying, we have
V1+ [f,(x, y)]² + [f,(x, v)]²
=](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F67afbb67-017a-4ebc-bbcb-7bf247bdda00%2F2a579450-ca15-4bf7-8b8e-aae9581e9b45%2Fgljrje1_processed.png&w=3840&q=75)
Transcribed Image Text:Tutorial Exercise
Find the area of the surface.
The portion of the cone z = 7
x² + y2 inside the cylinder x² + y2 = 9
Step 1
The definition of the surface area says if f and its first partial derivatives are continuous on the closed interval R in the xy-plane, then the area of the surface S given by z = f(x, y) over R is
S =
ds
= / | V1+ [f,(x, y)1² + [f,cx, v)]? ds.
We are asked to find the area of the portion of the cone z = 7VX2 + y² inside the cylinderx? + y2 = 9.
Step 2
To findf, (x, y), partially differentiate f(x, y) with respect to x.
x2
+ y
=
дх
7x
Vx² + y²
Similarly, find f,(x, y).
F,lx, y) = (7V;
x²
y2
ду
7
Therefore,
49x2
+
49
V1 + [f,(x, y)]² + [f,(x, y)]²
1 +
x2 + y2
x² + y2
Simplifying, we have
V1+ [f,(x, y)]² + [f,(x, v)]²
=
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