Textbook Question
Let R be the region bounded by the following curves. Find the volume of the solid generated when R is revolved about the given line.
x=2−secy,x=2,y=π/3, and y=0; about x=2
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9. Graphical Applications of Integrals
Introduction to Volume & Disk Method
2:36 minutesProblem 6.3.69a
Textbook Question
A right circular cylinder with height R and radius R has a volume of VC=πR^3 (height = radius).
a. Find the volume of the cone that is inscribed in the cylinder with the same base as the cylinder and height R. Express the volume in terms of VC.
Verified step by step guidance1
Identify the given dimensions: The cylinder has height \(R\) and radius \(R\), so its volume is given by \(V_C = \pi R^3\).
Recall the formula for the volume of a cone: \(V_{cone} = \frac{1}{3} \pi r^2 h\), where \(r\) is the radius of the base and \(h\) is the height.
Since the cone is inscribed in the cylinder with the same base and height, set \(r = R\) and \(h = R\) in the cone volume formula.
Substitute these values into the cone volume formula to get \(V_{cone} = \frac{1}{3} \pi R^2 R = \frac{1}{3} \pi R^3\).
Express the cone volume in terms of the cylinder volume \(V_C\): since \(V_C = \pi R^3\), then \(V_{cone} = \frac{1}{3} V_C\).
Verified video answer for a similar problem:This video solution was recommended by our tutors as helpful for the problem above
Video duration:
2mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Volume of a Cylinder
The volume of a right circular cylinder is calculated by multiplying the area of its circular base by its height. Specifically, the formula is V = πr²h, where r is the radius and h is the height. In this problem, both radius and height are equal to R, so the volume simplifies to V = πR³.
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Volume of a Cone
The volume of a cone is one-third the volume of a cylinder with the same base and height. The formula is V = (1/3)πr²h, where r is the radius of the base and h is the height. This relationship is essential for finding the cone's volume inscribed in the cylinder.
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Expressing Volume in Terms of Another Variable
Expressing one volume in terms of another involves substituting known values and simplifying. Here, the cone's volume should be expressed in terms of VC, the cylinder's volume, by using the relationship between their volumes and the given dimensions.
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