Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

9. Graphical Applications of Integrals

Introduction to Volume & Disk Method

Calculus

9. Graphical Applications of Integrals

Introduction to Volume & Disk Method

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2:36 minutes

Problem 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 guidance

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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Key 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.

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.

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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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.

Key Concepts

Volume of a Cylinder

Volume of a Cone

Expressing Volume in Terms of Another Variable

Watch next

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.