Table of contents

0. Functions7h 55m

Introduction to Functions
18m

Piecewise Functions
10m

Properties of Functions
9m

Common Functions
1h 8m

Transformations
5m

Combining Functions
27m

Exponent rules
32m

Exponential Functions
28m

Logarithmic Functions
24m

Properties of Logarithms
36m

Exponential & Logarithmic Equations
35m

Introduction to Trigonometric Functions
38m

Graphs of Trigonometric Functions
44m

Trigonometric Identities
47m

Inverse Trigonometric Functions
48m

1. Limits and Continuity2h 2m

Introduction to Limits
41m

Finding Limits Algebraically
40m

Continuity
40m

2. Intro to Derivatives1h 33m

Tangent Lines and Derivatives
30m

Derivatives as Functions
14m

Basic Graphing of the Derivative
23m

Differentiability
26m

3. Techniques of Differentiation3h 18m

Basic Rules of Differentiation
39m

Higher Order Derivatives
12m

Product and Quotient Rules
55m

Derivatives of Trig Functions
40m

The Chain Rule
49m

4. Applications of Derivatives2h 38m

Motion Analysis
36m

Implicit Differentiation
27m

Related Rates
59m

Linearization
13m

Differentials
21m

5. Graphical Applications of Derivatives6h 2m

Intro to Extrema
23m

Finding Global Extrema
50m

The First Derivative Test
1h 5m

Concavity
1h 1m

The Second Derivative Test
31m

Curve Sketching
44m

Applied Optimization
1h 25m

6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m

Derivatives of Exponential & Logarithmic Functions
1h 52m

Logarithmic Differentiation
20m

Derivatives of Inverse Trigonometric Functions
24m

7. Antiderivatives & Indefinite Integrals1h 26m

Antiderivatives
17m

Indefinite Integrals
25m

Integrals of Trig Functions
15m

Initial Value Problems
27m

8. Definite Integrals4h 44m

Estimating Area with Finite Sums
42m

Riemann Sums
1h 21m

Introduction to Definite Integrals
22m

Fundamental Theorem of Calculus
37m

Average Value of a Function
21m

Substitution
1h 19m

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 31m

Integrals of Exponential Functions
40m

Integrals Involving Logarithmic Functions
58m

Integrals Involving Inverse Trigonometric Functions
52m

12. Techniques of Integration7h 41m

Integration by Parts
2h 32m

Partial Fractions
4h 29m

Improper Integrals
39m

13. Intro to Differential Equations2h 55m

Basics of Differential Equations
48m

Slope Fields
19m

Euler's Method
22m

Separable Differential Equations
1h 25m

14. Sequences & Series5h 36m

Sequences
1h 22m

Review of Factorials
11m

Series
44m

Convergence Tests
3h 17m

15. Power Series2h 19m

Introduction to Power Series
1h 9m

Taylor Series & Taylor Polynomials
1h 10m

16. Parametric Equations & Polar Coordinates7h 58m

Parametric Equations
1h 6m

Calculus with Parametric Curves
1h 13m

Polar Coordinates
2h 5m

Calculus in Polar Coordinates
55m

Conic Sections
2h 36m

9. Graphical Applications of Integrals

Introduction to Volume & Disk Method

Calculus

9. Graphical Applications of Integrals

Introduction to Volume & Disk Method

Previous problemNext problem

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.