Ch. 6 - Applications of Definite Integrals

Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook

Hass 15th Edition

Ch. 6 - Applications of Definite Integrals

Problem 6.PE.7c

Chapter 6, Problem 6.PE.7c

Volumes
Find the volume of the solid generated by revolving the region bounded by the x-axis, the curve y = 3x⁴ , and the lines x = 1 and x = ―1 about
c. the line x = 1

Verified step by step guidance

Identify the region bounded by the curves: the x-axis (y = 0), the curve y = 3x^{4}, and the vertical lines x = -1 and x = 1.

Since the solid is generated by revolving the region about the vertical line x = 1, use the method of cylindrical shells, which is suitable for rotation around vertical lines other than the y-axis.

Set up the volume integral using the shell method formula: \[ V = 2\pi \int_{a}^{b} (\text{radius})(\text{height}) \, dx \] where the radius is the distance from the shell to the axis of rotation, and the height is the function value.

Determine the radius of a shell at position x: since the axis of rotation is x = 1, the radius is \(|1 - x|\). The height of the shell is the value of the function \(y = 3x^{4}\).

Set the limits of integration from x = -1 to x = 1, and write the integral explicitly as: \[ V = 2\pi \int_{-1}^{1} (1 - x)(3x^{4}) \, dx \] Note that \(1 - x\) is used because for x in [-1,1], \(1 - x\) is positive.

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Key Concepts

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

Volume of Solids of Revolution

This concept involves finding the volume of a 3D solid formed by rotating a 2D region around a given axis. Common methods include the disk/washer method and the shell method, which use integration to sum infinitesimal volumes. Choosing the appropriate method depends on the axis of rotation and the shape of the region.

Shell Method for Volumes

The shell method calculates volume by integrating cylindrical shells formed when revolving a region around a vertical or horizontal line. Each shell's volume is approximated by 2π(radius)(height)(thickness). This method is especially useful when rotating around vertical lines like x = 1, where integrating with respect to y or x simplifies the problem.

Setting up Integration Limits and Radius

Accurately determining the limits of integration and the radius of rotation is crucial. For rotation about x = 1, the radius is the horizontal distance from a point x to the line x = 1, given by |1 - x|. The limits come from the given bounds on x, here from -1 to 1, defining the region to be revolved.

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Related Practice

Textbook Question

Areas of Surfaces of Revolution

In Exercises 23–26, find the areas of the surfaces generated by revolving the curves about the given axes.

_______

y = √4y ― y² , 1 ≤ y ≤ 2 ; y-axis

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Textbook Question

Volumes

Find the volume of the solid generated by revolving the region bounded by the curve y = sin x and the lines x = 0, x = π and y = 2 about the line y = 2.

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Textbook Question

Volumes

Find the volume of the solid generated by revolving the region bounded on the left by the parabola x = y² + 1 and on the right by the line x = 5 about

a. the x-axis

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Textbook Question

Volumes

Find the volumes of the solids in Exercises 1–18.

The solid lies between planes perpendicular to the x-axis at x = π/4 and x = 5π/4. The cross-sections between these planes are circular disks whose diameters run from the curve y = 2 cos x to the curve y = 2 sin x.

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Textbook Question

Volumes

Find the volume of the solid generated by revolving the region between the x-axis and curve y = x² ―2x about

b. the line y = ―1

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Textbook Question

Find the lengths of the curves in Exercises 19–22.

y = (5/12) x⁶/⁵ ― (5/8)x⁴/⁵ , 1 ≤ x ≤ 32

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