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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2:09 minutes

Problem 6.3.69b

Textbook Question

A right circular cylinder with height R and radius R has a volume of VC=πR^3 (height = radius).

b. Find the volume of the hemisphere that is inscribed in the cylinder with the same base as the cylinder. Express the volume in terms of VC.

Verified step by step guidance

Identify the dimensions of the hemisphere inscribed in the cylinder. Since the hemisphere shares the same base as the cylinder, its radius is equal to the radius of the cylinder, which is $R$.

Recall the formula for the volume of a hemisphere: $V_H = \frac{2}{3} \pi r^3$. Substitute $r = R$ to get $V_H = \frac{2}{3} \pi R^3$.

Recall the volume of the cylinder given as $V_C = \pi R^3$ (since height = radius = $R$).

Express the volume of the hemisphere $V_H$ in terms of the cylinder volume $V_C$ by dividing $V_H$ by $V_C$: $\frac{V_H}{V_C} = \frac{\frac{2}{3} \pi R^3}{\pi R^3}$.

Simplify the expression to find $V_H$ as a fraction of $V_C$, which will give the volume of the hemisphere in terms of the cylinder volume.

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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 base by its height. For a cylinder with radius R and height R, the volume is V = πR² × R = πR³. This formula provides the reference volume VC used in the problem.

Volume of a Hemisphere

A hemisphere is half of a sphere. The volume of a sphere with radius r is (4/3)πr³, so the volume of a hemisphere is half of that, (2/3)πr³. Understanding this formula is essential to find the volume of the hemisphere inscribed in the cylinder.

Inscribed Shapes and Radius Relationship

An inscribed hemisphere inside a cylinder with the same base means the hemisphere's radius equals the cylinder's radius R. Recognizing this relationship allows substitution of R into the hemisphere volume formula and expressing the result in terms of VC.

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A right circular cylinder with height R and radius R has a volume of VC=πR^3 (height = radius). b. Find the volume of the hemisphere that is inscribed in the cylinder with the same base as the cylinder. Express the volume in terms of VC.

Key Concepts

Volume of a Cylinder

Volume of a Hemisphere

Inscribed Shapes and Radius Relationship

Watch next

A right circular cylinder with height R and radius R has a volume of VC=πR^3 (height = radius).

b. Find the volume of the hemisphere that is inscribed in the cylinder with the same base as the cylinder. Express the volume in terms of VC.