Question

Centroid: Find the centroid of the region bounded by the x-axis, the curve y = csc x, and the lines x = π/6, x = 5π/6.

Accepted Answer

Identify the region bounded by the x-axis (y = 0), the curve y = \csc x, and the vertical lines x = \frac{\pi}{6} and x = \frac{5\pi}{6}. This region lies above the x-axis since \csc x is positive in this interval. Recall that the centroid (\bar{x}, \bar{y}) of a planar region bounded by curves can be found using the formulas:

$$\displaystyle \bar{x} = \frac{1}{A} \int_a^b x f(x) \, dx$$

$$\displaystyle \bar{y} = \frac{1}{2A} \int_a^b [f(x)]^2 \, dx$$

where $$A is $$the area of the region, $$f(x) is $$the upper boundary function, and $$[a,b] is $$the interval on the x-axis. Calculate the area $$A of $$the region using the integral:

$$\displaystyle A = \int_{\frac{\pi}{6}}^{\frac{5\pi}{6}} \csc x \, dx.

$$This integral represents the area under the curve y = \csc x from x = \frac{\pi}{6} to x = \frac{5\pi}{6}. Set up the integral for the x-coordinate of the centroid:

$$\displaystyle \bar{x} = \frac{1}{A} \int_{\frac{\pi}{6}}^{\frac{5\pi}{6}} x \csc x \, dx.

$$This integral weights each x-value by the height of the curve at that point. Set up the integral for the y-coordinate of the centroid:

$$\displaystyle \bar{y} = \frac{1}{2A} \int_{\frac{\pi}{6}}^{\frac{5\pi}{6}} (\csc x)^2$ \, dx.

$$This integral accounts for the vertical distribution of the area to find the average height.

Centroid: Find the centroid of the region bounded by the x-axis, the curve y = csc x, and the lines x = π/6, x = 5π/6.

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Key Concepts

Centroid of a Plane Region

Definite Integration for Area and Moments

Properties and Behavior of the Function y = csc x