Volumes by Cylindrical Shells2

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Mathematics
•
12th Grade
•
Hard
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1.
FLASHCARD QUESTION
Front
WHAT IS THE RADIUS OF A CYLINDRICAL SHELL?
Back
The radius of a cylindrical shell is the distance from the axis of rotation to the shell. It can be expressed as a function of the variable, often denoted as r(x). For example, if the shell is defined by the equation r = 3 - x, then the radius decreases linearly as x increases.
Tags
CCSS.8.G.C.9
CCSS.HSG.GMD.A.3
2.
FLASHCARD QUESTION
Front
HOW DO YOU DETERMINE THE HEIGHT OF A CYLINDRICAL SHELL?
Back
The height of a cylindrical shell is determined by the difference between the upper and lower functions that define the region being revolved. If the region is bounded by y = f(x) and y = g(x), the height h is given by h = f(x) - g(x).
3.
FLASHCARD QUESTION
Front
WHAT IS THE FORMULA FOR THE VOLUME OF A CYLINDRICAL SHELL?
Back
The volume V of a cylindrical shell is given by the formula: V = 2π ∫[a to b] (radius)(height) dx, where the radius and height are functions of x.
Tags
CCSS.8.G.C.9
CCSS.HSG.GMD.A.3
4.
FLASHCARD QUESTION
Front
WHAT DOES THE INTEGRAL IN THE VOLUME FORMULA REPRESENT?
Back
The integral in the volume formula represents the accumulation of the volumes of infinitesimally thin cylindrical shells from x = a to x = b.
Tags
CCSS.8.G.C.9
CCSS.HSG.GMD.A.3
5.
FLASHCARD QUESTION
Front
HOW DOES THE CHOICE OF AXIS OF ROTATION AFFECT THE VOLUME CALCULATION?
Back
The choice of axis of rotation affects the expressions for the radius and height of the shells, which in turn influences the volume calculation. Different axes may require different functions for radius and height.
Tags
CCSS.8.G.C.9
CCSS.HSG.GMD.A.3
6.
FLASHCARD QUESTION
Front
WHAT IS THE SIGNIFICANCE OF THE LIMITS OF INTEGRATION IN THE VOLUME FORMULA?
Back
The limits of integration in the volume formula define the interval over which the shells are being summed. They correspond to the bounds of the region being revolved.
7.
FLASHCARD QUESTION
Front
HOW DO YOU SET UP AN INTEGRAL FOR VOLUME USING CYLINDRICAL SHELLS?
Back
To set up an integral for volume using cylindrical shells, identify the radius and height as functions of x, determine the limits of integration based on the region, and apply the formula V = 2π ∫[a to b] (radius)(height) dx.
Tags
CCSS.8.G.C.9
CCSS.HSG.GMD.A.3
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