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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2:34 minutes

Problem 8.3.65

Textbook Question

65. Volume Find the volume of the solid generated when the region bounded by y = sin²(x) * cos^(3/2)(x) and the x-axis on the interval [0, π/2] is revolved about the x-axis.

Verified step by step guidance

Step 1: Recognize that the problem involves finding the volume of a solid of revolution. The formula for the volume when a region is revolved about the x-axis is given by:

V = π ∫[a,b] f(x)² dx

, where f(x) is the function describing the region and [a, b] is the interval of integration.

Step 2: Identify the function and interval from the problem. Here, the function is

f(x) = sin²(x) * cos^(3/2)(x)

, and the interval is

[0, π/2]

. Substitute these into the volume formula.

Step 3: Set up the integral for the volume. The integral becomes:

V = π ∫[0, π/2] (sin²(x) * cos^(3/2)(x))² dx

. Simplify the integrand by squaring the function:

(sin²(x) * cos^(3/2)(x))² = sin⁴(x) * cos³(x)

Step 4: Rewrite the integral with the simplified integrand:

V = π ∫[0, π/2] sin⁴(x) * cos³(x) dx

. To solve this integral, consider using trigonometric identities or substitution methods. For example, you might use the identity

sin²(x) = 1 - cos²(x)

to simplify further.

Step 5: Apply appropriate integration techniques, such as substitution or reduction formulas, to evaluate the integral. Once the integral is solved, multiply the result by π to find the volume. Ensure all steps are carefully followed to handle the trigonometric powers correctly.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Volume of Revolution

The volume of revolution refers to the volume of a solid formed by rotating a two-dimensional shape around an axis. In calculus, this is typically calculated using methods such as the disk method or the washer method, which involve integrating the area of circular cross-sections perpendicular to the axis of rotation.

Definite Integral

A definite integral calculates the accumulation of quantities, such as area or volume, over a specific interval. In this context, it is used to find the volume of the solid generated by revolving the given function around the x-axis, integrating from the lower to the upper bounds of the interval, [0, π/2].

Trigonometric Functions

Trigonometric functions, such as sine and cosine, are fundamental in calculus, especially when dealing with periodic phenomena. The function y = sin²(x) * cos^(3/2)(x) combines these functions, and understanding their behavior and properties is essential for evaluating the integral and determining the volume of the solid formed.

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Find the volume of the solid obtained by rotating the region bounded by $y = x + 4$ , $y equals 0$ , $x equals 1$ & $x equals 5$ .

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Find the volume of the solid formed by revolving the area bounded by $f (x) = x^{2}$ from $x equals 0$ to $x equals 3$ and the y-axis around the y-axis.