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Ch. 8 - Techniques of Integration
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 8 - Techniques of IntegrationProblem 8.3.72
Chapter 8, Problem 8.3.72

Center of gravity: Find the center of gravity of the region bounded by the x-axis, the curve y = sec x, and the lines x = -pi/4 and x = pi/4.

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Identify the region bounded by the x-axis, the curve \(y = \sec x\), and the vertical lines \(x = -\frac{\pi}{4}\) and \(x = \frac{\pi}{4}\). This region lies above the x-axis since \(\sec x\) is positive in this interval.
Recall that the center of gravity (centroid) \((\bar{x}, \bar{y})\) of a planar region can be found using the formulas: \(\displaystyle \bar{x} = \frac{1}{A} \int_a^b x f(x) \, dx\) and \(\displaystyle \bar{y} = \frac{1}{2A} \int_a^b [f(x)]^2 \, dx\), where \(A\) is the area of the region bounded by \(y = f(x)\) and the x-axis from \(x = a\) to \(x = b\).
Calculate the area \(A\) of the region using the integral: \(\displaystyle A = \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \sec x \, dx\). This integral represents the total area under the curve \(y = \sec x\) between the given limits.
Compute the \(x\)-coordinate of the centroid using: \(\displaystyle \bar{x} = \frac{1}{A} \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} x \sec x \, dx\). Note that because the function \(\sec x\) is even and \(x\) is odd, this integral will have symmetry properties that can simplify the calculation.
Compute the \(y\)-coordinate of the centroid using: \(\displaystyle \bar{y} = \frac{1}{2A} \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} (\sec x)^2 \, dx\). This integral gives the moment about the x-axis, which when divided by twice the area, yields the vertical coordinate of the center of gravity.

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Key Concepts

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

Center of Gravity (Centroid) of a Region

The center of gravity or centroid of a planar region is the point where the region's area can be considered to be concentrated. It is found by calculating the coordinates (x̄, ȳ), which are the weighted averages of the x and y coordinates over the area. For regions bounded by curves, these coordinates are determined using definite integrals.
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Definite Integrals for Area and Moments

To find the centroid, you need to compute the area of the region and the moments about the coordinate axes. The area is found by integrating the function defining the curve minus the x-axis over the given interval. Moments involve integrating x·f(x) and (1/2)·[f(x)]² to find the weighted sums needed for centroid coordinates.
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Properties and Behavior of the Secant Function

The secant function, sec x = 1/cos x, is defined where cos x ≠ 0 and is symmetric about the y-axis. Understanding its shape and values between -π/4 and π/4 is crucial for setting up the integral limits and ensuring the region is properly bounded by the x-axis and the curve y = sec x.
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