MAR P # Video Example) EXAMPLE 8 Find the volume of a pyramid whose base is a square with side L and whose height is h. SOLUTION We place the origin O at the vertex of the pyramid and the x-axis along its central axis as in the top figure. Any plane P that passes through x and is perpendicular to the x-axis intersects the pyramid in a square with side of length s, say. We can express s in terms of x by observing from the similar triangles the bottom figure that and so s= . [Another method is to observe that the line OP has slope L/(2h) and so its equation is x y = Lx/(2h).] Thus the cross-sectional area is A(x) = s² = The pyramid lies between x = 0 and x = , so its volume is V = ["A(X) <- 6² ²/1/² * x² dx = - A(x) dx = [
MAR P # Video Example) EXAMPLE 8 Find the volume of a pyramid whose base is a square with side L and whose height is h. SOLUTION We place the origin O at the vertex of the pyramid and the x-axis along its central axis as in the top figure. Any plane P that passes through x and is perpendicular to the x-axis intersects the pyramid in a square with side of length s, say. We can express s in terms of x by observing from the similar triangles the bottom figure that and so s= . [Another method is to observe that the line OP has slope L/(2h) and so its equation is x y = Lx/(2h).] Thus the cross-sectional area is A(x) = s² = The pyramid lies between x = 0 and x = , so its volume is V = ["A(X) <- 6² ²/1/² * x² dx = - A(x) dx = [
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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please check screenshot
![Video Example)
h
EXAMPLE 8 Find the volume of a pyramid whose base is a square with side L and whose height is h.
SOLUTION We place the origin O at the vertex of the pyramid and the x-axis along its central axis as in the top figure. Any
plane P that passes through x and is perpendicular to the x-axis intersects the pyramid in a square with side of length s, say.
We can express s in terms of x by observing from the similar triangles in the bottom figure that
s/2
L/2
and so s
[Another method is to observe that the line OP has slope L/(2h) and so its equation is
x
y = Lx/(2h).] Thus the cross-sectional area is
A(x) = s² =
The pyramid lies between x = 0 and x =
so its volume is
,2
V=
A(x) dx =
- 1²/²1/²2 × ²
x² dx =
11-[](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc2e97f32-5988-4deb-9547-eb6ce37eb1f3%2F1f3ff197-b8e8-4aa6-8880-9ee1b53c4f2d%2Fu5tqrei_processed.png&w=3840&q=75)
Transcribed Image Text:Video Example)
h
EXAMPLE 8 Find the volume of a pyramid whose base is a square with side L and whose height is h.
SOLUTION We place the origin O at the vertex of the pyramid and the x-axis along its central axis as in the top figure. Any
plane P that passes through x and is perpendicular to the x-axis intersects the pyramid in a square with side of length s, say.
We can express s in terms of x by observing from the similar triangles in the bottom figure that
s/2
L/2
and so s
[Another method is to observe that the line OP has slope L/(2h) and so its equation is
x
y = Lx/(2h).] Thus the cross-sectional area is
A(x) = s² =
The pyramid lies between x = 0 and x =
so its volume is
,2
V=
A(x) dx =
- 1²/²1/²2 × ²
x² dx =
11-[
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