Problem 19 - Calculus Early Transcendentals

Question

Find the mass and centroid (center of mass) of the following thin plates, assuming constant density. Sketch the region corresponding to the plate and indicate the location of the center of mass. Use symmetry when possible to simplify your work. The region bounded by \(y=\ln x,\) the \(x\) -axis, and \(x=e\)

Step-by-Step Solution

Answer

Based on the step-by-step solution above, provide a short answer to describe the centroid of the thin plate: Since the mass of the thin plate is found to be 0, it implies that there is no laminar material in the plate. Therefore, there is no centroid for this thin plate.

Step 1: Define the region and set up mass integrals

The region we are analyzing is bounded by y = ln(x), the x-axis, and x = e. Since density is constant, we need to find the area of this region to get the mass of the plate. We'll perform the following integrals to get the mass: $$m = \int_{1}^{e} y \, dx = \int_{1}^{e} \ln x \, dx$$

Step 2: Compute the mass integral

Using integration by parts, let \(u = \ln x\) and dv = \(dx\). Then, \(du = \frac{1}{x} dx\) and \(v = x\). Now we compute the integral: $$m = \int_{1}^{e} \ln x \, dx = \left[ x \ln x - x \right]_1^e = e(1) - e - (1 - 1) = e - e = 0$$ Since the mass of the plate is 0, there is no laminar material and thus, there is no centroid. However, we will continue the procedure of finding centroid under normal circumstances.

Step 3: Centroid coordinates formula

Under regular conditions, we would find the coordinates \((\bar{x}, \bar{y})\) of the centroid using the following formula: $$\bar{x} = \frac{1}{m} \int_{1}^{e} x \ln x \, dx$$ $$\bar{y} = \frac{1}{2m} \int_{1}^{e} (\ln x)^2 dx$$

Step 4: Compute the centroid coordinates integral

To maintain consistency with the solution, we would proceed with the following integrals. To find \(\bar{x}\), we again use integration by parts: Let \(u = \ln x\) and \(dv = x dx\). Then \(du = \frac{1}{x} dx\) and \(v = \frac{1}{2} x^{2}\). We compute the integral: $$\int_{1}^{e} x \ln x \, dx = \left[ \frac{1}{2} x^{2} \ln x - \frac{1}{4} x^{2} \right]_1^e = \frac{1}{2} e^{2}(1) - \frac{1}{4} e^{2} - (\frac{1}{2}(1) - \frac{1}{4})$$ To find \(\bar{y}\), we use the integral: $$\int_{1}^{e} (\ln x)^2 dx$$ This step would require integration by a more advanced technique such as the gamma function, which is beyond the scope of this problem. However, we have found that the mass of the plate is 0, thus there is no centroid.