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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1:31 minutes

Problem 6.4.8b

Textbook Question

6–8. Let R be the region bounded by the curves y = 2−√x,y=2, and x=4 in the first quadrant.

Suppose the shell method is used to determine the volume of the solid generated by revolving R about the line x=4.

b. What is the height of a cylindrical shell at a point x in [0, 4]?

Verified step by step guidance

Step 1: Understand the problem. The region R is bounded by the curves y = 2−√x, y = 2, and x = 4 in the first quadrant. We are tasked with finding the height of a cylindrical shell at a point x in [0, 4] when the region is revolved about the line x = 4 using the shell method.

Step 2: Recall the shell method formula. The height of a cylindrical shell is determined by the vertical distance between the top curve and the bottom curve at a given x-coordinate. In this case, the top curve is y = 2 and the bottom curve is y = 2−√x.

Step 3: Calculate the height of the shell. The height is given by the difference between the top curve and the bottom curve: height = y_top − y_bottom. Substituting the equations, height = 2 − (2−√x).

Step 4: Simplify the expression for the height. Perform the subtraction: height = 2 − 2 + √x, which simplifies to height = √x.

Step 5: Conclude that the height of the cylindrical shell at a point x in [0, 4] is √x. This height will be used in the shell method formula to compute the volume of the solid.

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Key Concepts

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

Cylindrical Shell Method

The cylindrical shell method is a technique for finding the volume of a solid of revolution. It involves slicing the solid into thin cylindrical shells, which are then integrated to find the total volume. When revolving around a vertical line, the height of each shell is determined by the function defining the region, and the radius is the distance from the axis of rotation.

Height of a Shell

In the context of the shell method, the height of a cylindrical shell at a point x is given by the difference between the upper and lower bounding functions of the region being revolved. For the given region R, the height is calculated as the vertical distance between the line y = 2 and the curve y = 2 - √x, specifically expressed as h(x) = 2 - (2 - √x) = √x.

Bounded Region

A bounded region in calculus refers to a specific area enclosed by curves or lines on a graph. In this case, region R is bounded by the curves y = 2, y = 2 - √x, and the vertical line x = 4. Understanding the boundaries is crucial for accurately applying integration techniques to find volumes or areas related to the region.

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Textbook Question

A surface is generated by revolving the line f(x)=2−x, for 0≤x≤2, about the x-axis. Find the area of the resulting surface in the following ways.

a. Using calculus

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Textbook Question

Let R be the region in the first quadrant bounded above by the curve y=2−x² and bounded below by the line y=x. Suppose the shell method is used to determine the volume of the solid generated by revolving R about the y-axis.

b. What is the height of a cylindrical shell at a point x in [0, 2]?

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Textbook Question

9-34. Shell method Let R be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when R is revolved about indicated axis.

y = (1+x²)^−1,y = 0,x = 0, and x = 2; about the y-axis

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Textbook Question

6–8. Let R be the region bounded by the curves y = 2−√x,y=2, and x=4 in the first quadrant.

Suppose the shell method is used to determine the volume of the solid generated by revolving R about the line x=4.

a. What is the radius of a cylindrical shell at a point x in [0, 4]?

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Textbook Question

6–8. Let R be the region bounded by the curves y = 2−√x,y=2, and x=4 in the first quadrant.

Suppose the shell method is used to determine the volume of the solid generated by revolving R about the line x=4.

c. Write an integral for the volume of the solid using the shell method.

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Textbook Question

9-34. Shell method Let R be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when R is revolved about indicated axis.

x = 4 / y + y³,x = 1/√3, and y=1; about the x-axis

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Textbook Question

Use the region R that is bounded by the graphs of y=1+√x,x=4, and y=1 complete the exercises.

Region R is revolved about the x-axis to form a solid of revolution whose cross sections are washers.

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

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Textbook Question

Use the region R that is bounded by the graphs of y=1+√x,x=4, and y=1 complete the exercises.

Region R is revolved about the x-axis to form a solid of revolution whose cross sections are washers.

d. Write an integral for the volume of the solid.

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6–8. Let R be the region bounded by the curves y = 2−√x,y=2, and x=4 in the first quadrant.Suppose the shell method is used to determine the volume of the solid generated by revolving R about the line x=4.b. What is the height of a cylindrical shell at a point x in [0, 4]?

Key Concepts

Cylindrical Shell Method

Height of a Shell

Bounded Region

Watch next

6–8. Let R be the region bounded by the curves y = 2−√x,y=2, and x=4 in the first quadrant.

Suppose the shell method is used to determine the volume of the solid generated by revolving R about the line x=4.

b. What is the height of a cylindrical shell at a point x in [0, 4]?