Ch. 8 - Integration Techniques

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 8 - Integration Techniques

Problem 8.R.101

Chapter 8, Problem 8.R.101

101. Comparing volumes Let R be the region bounded by the graph of y = sin(x) and the x-axis on the interval [0, π]. Which is greater, the volume of the solid generated when R is revolved about the x-axis or about the y-axis?

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First, identify the region R bounded by the curve \(y = \sin(x)\) and the x-axis on the interval \([0, \pi]\). This means the region lies between \(y = 0\) and \(y = \sin(x)\) for \(x\) in \([0, \pi]\).

To find the volume when R is revolved about the x-axis, use the disk method. The volume \(V_x\) is given by the integral \(V_x = \pi \int_0^{\pi} (\sin(x))^2 \, dx\) because the radius of each disk is \(\sin(x)\).

To find the volume when R is revolved about the y-axis, use the shell method. The volume \(V_y\) is given by \(V_y = 2\pi \int_0^{\pi} x \sin(x) \, dx\), where \(x\) is the radius of the shell and \(\sin(x)\) is the height.

Set up both integrals explicitly: \(V_x = \pi \int_0^{\pi} \sin^2(x) \, dx\) \(V_y = 2\pi \int_0^{\pi} x \sin(x) \, dx\)

Evaluate both integrals separately (using appropriate integration techniques such as power-reduction for \(\sin^2(x)\) and integration by parts for \(x \sin(x)\)), then compare the two volumes to determine which is greater.

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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 for rotation about the x-axis and the shell method for rotation about the y-axis. Understanding these methods helps set up the correct integrals for volume calculation.

Disk/Washer Method

Used when revolving a region around the x-axis, this method slices the solid perpendicular to the axis of rotation, creating circular disks or washers. The volume is found by integrating the area of these cross-sectional disks along the interval, typically using the formula π∫[f(x)]² dx.

Shell Method

This method is useful for revolving a region around the y-axis. It involves slicing the region parallel to the axis of rotation, forming cylindrical shells. The volume is calculated by integrating the lateral surface area of these shells, using the formula 2π∫(radius)(height) dx.

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Related Practice

Textbook Question

109. Average velocity Find the average velocity of a projectile whose velocity over the interval 0 ≤ t ≤ π is given by

v(t) = 10 * sin(3t).

Textbook Question

89–91. Comparison Test Determine whether the following integrals converge or diverge.

89. ∫ (from 1 to ∞) dx/(x⁵ + x⁴ + x³ + 1)

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Textbook Question

82-88. Improper integrals Evaluate the following integrals or show that the integral diverges.

86. ∫ (from -∞ to ∞) x³/(1 + x⁸) dx

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Textbook Question

125. Wallis products Complete the following steps to prove a well-known formula discovered by the 17th-century English mathematician John Wallis.

a. Use a reduction formula to show that ∫ from 0 to π of (sin^m x) dx = (m − 1)/m × ∫ from 0 to π of (sin^(m−2) x) dx, for any integer m ≥ 2.

Textbook Question

Explain why or why not. Determine whether the following statements are true and give an explanation or counterexample.

e. The best approach to evaluating ∫(x³ + 1)/(3x²) dx is to use the change of variables u = x³ + 1.

Textbook Question

102–105. Volumes The region R is bounded by the curve y = ln(x) and the x-axis on the interval [1, e]. Find the volume of the solid generated when R is revolved in the following ways.

102. About the y-axis