Ch. 8 - Techniques of Integration

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

Hass 15th Edition

Ch. 8 - Techniques of Integration

Problem 8.2.64b

Chapter 8, Problem 8.2.64b

Consider the region bounded by the graphs of
y = arctan(x), y = 0, and x = 1.
b. Find the volume of the solid formed by revolving this region about the y-axis.

Verified step by step guidance

First, identify the region bounded by the curves: \(y = \arctan(x)\), \(y = 0\), and the vertical line \(x = 1\). This region lies between \(x=0\) and \(x=1\) since \(y=0\) corresponds to the \(x\)-axis and \(y=\arctan(x)\) is above it for \(x > 0\).

Since the solid is formed by revolving the region about the \(y\)-axis, consider using the method of cylindrical shells. The formula for the volume using cylindrical shells when revolving around the \(y\)-axis is: \(V = \int_a^b 2\pi \cdot (\text{radius}) \cdot (\text{height}) \, dx\)

In this problem, the radius of a shell at position \(x\) is the distance from the \(y\)-axis, which is simply \(x\). The height of the shell is the vertical distance between the curves, which is \(\arctan(x) - 0 = \arctan(x)\).

Set up the integral for the volume as: \(V = \int_0^1 2\pi x \cdot \arctan(x) \, dx\)

To find the volume, you would evaluate the integral \(\int_0^1 x \arctan(x) \, dx\) and then multiply the result by \(2\pi\). Integration by parts is a suitable method to solve this integral.

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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 an 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 given boundaries.

Shell Method

The shell method calculates volume by integrating cylindrical shells formed by revolving vertical slices around a vertical axis. Each shell's volume is approximated by 2π(radius)(height)(thickness). This method is especially useful when revolving around the y-axis and the function is given in terms of x.

Inverse Trigonometric Functions and Their Properties

Understanding the function y = arctan(x) is crucial, including its domain, range, and behavior. Arctan(x) is the inverse of the tangent function, continuous and increasing for all real x, with horizontal asymptotes at ±π/2. This helps in setting integration limits and interpreting the region bounded by the curves.

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Derivatives of Other Inverse Trigonometric Functions

Related Practice

Textbook Question

88. The region in Exercise 87 is revolved about the x-axis to generate a solid.

b. Show that the inner and outer surfaces of the solid have infinite area.

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Textbook Question

In Exercises 11–22, estimate the minimum number of subintervals needed to approximate the integrals with an error of magnitude less than 10^-4 by (b) Simpson’s Rule. (The integrals in Exercises 11–18 are the integrals from Exercises 1–8.)

∫ from 0 to 2 of sin(x + 1) dx

Textbook Question

90. Consider the infinite region in the first quadrant bounded by the graphs of

y = 1 / √x, y = 0, x = 0, and x = 1.

b. Find the volume of the solid formed by revolving the region (i) about the x-axis

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Textbook Question

∫ from 1 to 2 of x dx

Textbook Question

Centroid:

Find the centroid of the region cut from the first quadrant by the curve

y = 1/√(x + 1) and the line x = 3.