Cylinders Let S be the solid in ℝ 3 between the cylinder z = f ( x ) and the region, where f ( x ) ≥ 0 on R. Explain why ∫ c d ∫ a b f ( x ) d x d y equals the area of the constant cross section of S multiplied by (d – c), which is the volume of S.
Cylinders Let S be the solid in ℝ 3 between the cylinder z = f ( x ) and the region, where f ( x ) ≥ 0 on R. Explain why ∫ c d ∫ a b f ( x ) d x d y equals the area of the constant cross section of S multiplied by (d – c), which is the volume of S.
Solution Summary: The author explains that the volume of the solid is equal to the (d-c) times the area of a constant cross section.
Cylinders Let S be the solid in
ℝ
3
between the cylinder z = f(x) and the region, where f(x) ≥ 0 on R. Explain why
∫
c
d
∫
a
b
f
(
x
)
d
x
d
y
equals the area of the constant cross section of S multiplied by (d – c), which is the volume of S.
Solids of revolution Let R be the region bounded by y = ln x, the x-axis, and the line x = e as shown. Find the volume of the solid that is generated when the region R is revolved about the x-axis.
find the volume
The base of a solid is the region bounded by the graphs of y = 3x, y = 6, and x = 0. The cross-sections perpendicular to the x-axis are a. rectangles of height 10. b. rectangles of perimeter 20.
Solids of revolution Let R be the region bounded by the following curves. Find the volume of the solid generated when R is revolved about the given axis.
y=e−x, y=0, x=0, and x=ln(4); about the x-axis
Calculus, Single Variable: Early Transcendentals (3rd Edition)
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.
Area Between The Curve Problem No 1 - Applications Of Definite Integration - Diploma Maths II; Author: Ekeeda;https://www.youtube.com/watch?v=q3ZU0GnGaxA;License: Standard YouTube License, CC-BY