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 8.2.69

Chapter 8, Problem 8.2.69

69. Comparing volumes Let R be the region bounded by y = sin x and the x-axis on the interval [0, π]. Which is greater, the volume when R is revolved about the x-axis, or the volume when R is revolved 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 the integral: $V_y = 2\pi \int_0^{\pi} x \cdot \sin x \, dx$ where $x$ is the radius of the shell and $\sin x$ is the height.

Set up both integrals explicitly and prepare to evaluate them (though we won't compute the final values here). This involves recognizing the integrals: $\int_0^{\pi} (\sin x)^2 \, dx$ and $\int_0^{\pi} x \sin x \, dx$.

After evaluating both integrals, compare the numerical values of $V_x$ and $V_y$ to determine which volume 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 generated when a region is revolved around an axis. Common methods include the disk/washer method for revolution about the x-axis and the shell method for revolution 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 revolution, creating circular disks or washers. The volume is found by integrating the area of these cross-sectional disks along the interval of interest.

Shell Method

This method is useful for revolving a region around the y-axis. It involves slicing the region parallel to the axis of revolution, forming cylindrical shells. The volume is calculated by integrating the lateral surface area of these shells over the given interval.

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