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Ch. 6 - Applications of Integration
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 6 - Applications of IntegrationProblema 6.3.69a
Capitolo 6, Problema 6.3.69a

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.

Guida verificata passo dopo passo
1
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 \(V_{cone} = \frac{1}{3} V_C\).

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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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04:48
Finding Volume Using Disks

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.
Video consigliato:
04:48
Finding Volume Using Disks

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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04:53
Finding Volume Using Disks Example 3
Pratica correlata
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Volume of a sphere Let R be the region bounded by the upper half of the circle x²+y² = r² and the x-axis. A sphere of radius r is obtained by revolving R about the x-axis.


a. Use the shell method to verify that the volume of a sphere of radius r is 4/3 πr³.

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13–16. Displacement from velocity Consider an object moving along a line with the given velocity v. Assume time t is measured in seconds and velocities have units of m/s.


a. Determine when the motion is in the positive direction and when it is in the negative direction. 


v(t) = 50e^−2t on [0, 4]

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Functions from arc length What differentiable functions have an arc length on the interval [a, b] given by the following integrals? Note that the answers are not unique. Give a family of functions that satisfy the conditions.

a. ∫a^b √1+16x⁴ dx

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Region R is revolved about the line y=1 to form a solid of revolution.


a. What is the radius of a cross section of the solid at a point x in [0, 4]?

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Consider the region R in the first quadrant bounded by y=x^1/n and y=x^n, where n>1 is a positive number.


a. Find the volume V(n) of the solid generated when R is revolved about the x-axis. Express your answer in terms of n.

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21–30. {Use of Tech} Arc length by calculator


a. Write and simplify the integral that gives the arc length of the following curves on the given interval. 

y = x³/3, for −1≤x≤1

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