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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1:59 minutes

Problem 6.4.8c

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.

Verified step by step guidance

Step 1: Understand the shell method. The shell method involves integrating the volume of cylindrical shells formed by revolving a region around a vertical or horizontal axis. The formula for the volume is: \( V = \int_{a}^{b} 2\pi \cdot \text{radius} \cdot \text{height} \cdot dx \).

Step 2: Identify the radius and height of the shells. Since the region R is revolved around the line \( x = 4 \), the radius of a shell is \( 4 - x \) (distance from the line \( x = 4 \) to a point \( x \) in the region). The height of the shell is given by the difference between the upper curve \( y = 2 \) and the lower curve \( y = 2 - \sqrt{x} \).

Step 3: Set up the integral bounds. The region R is bounded horizontally between \( x = 0 \) and \( x = 4 \). These will be the limits of integration.

Step 4: Write the integral expression for the volume. Using the shell method formula, the integral becomes: \( V = \int_{0}^{4} 2\pi (4 - x) \cdot [(2) - (2 - \sqrt{x})] \cdot dx \).

Step 5: Simplify the height expression inside the integral. The height \( (2) - (2 - \sqrt{x}) \) simplifies to \( \sqrt{x} \). Thus, the integral for the volume becomes: \( V = \int_{0}^{4} 2\pi (4 - x) \cdot \sqrt{x} \cdot dx \).

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Key Concepts

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

Region Bounded by Curves

In this problem, the region R is defined by the curves y = 2, y = 2 - √x, and the vertical line x = 4. Understanding how to identify and sketch the area bounded by these curves is crucial for visualizing the solid of revolution. The intersection points of these curves help determine the limits of integration when calculating volume.

Shell Method

The shell method is a technique for finding the volume of a solid of revolution. It involves slicing the solid into cylindrical shells and integrating the volume of these shells. When revolving around a vertical line, the formula for the volume V is given by V = 2π ∫ (radius)(height) dx, where the radius is the distance from the axis of rotation to the shell and height is the function value.

Setting Up the Integral

To write the integral for the volume using the shell method, one must determine the appropriate limits of integration and the expressions for the radius and height. In this case, the radius is (4 - x) and the height is (2 - (2 - √x)) = √x. The integral will be set up from x = 0 to x = 4, capturing the entire region R as it revolves around the line x = 4.

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Related Practice

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]?

views

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]?

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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]?

views

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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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 y-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 y in [1, 3]?

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

Key Concepts

Region Bounded by Curves

Shell Method

Setting Up the Integral

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