Question

Finding volumeLet R be the "triangular" region in the first quadrant that is bounded above by the line y = 1, below by the curve y = ln x, and on the left by the line x = 1.Find the volume of the solid generated by revolving R abouta. the x-axis.

Accepted Answer

First, identify the region R bounded by the curves: above by $$y = 1$$, below by $$y = \ln x$$, and on the left by $$x = 1. $$Since $$y = \ln x is $$defined for $$x \u003e 0$$, and the region is in the first quadrant, determine the right boundary by finding the $$x-$$value where $$y = 1$$ intersects $$y = \ln x. $$Solve $$\ln x = 1 to $$find this $$x-$$value. Set up the volume integral using the method of washers (disks with holes) since the region is revolved around the x-axis. The volume element is given by $$\pi 	imes (	ext{outer radius}^2 - 	ext{inner radius}^2) \, dx$$ where the outer radius is the distance from the x-axis to the upper curve and the inner radius is the distance from the x-axis to the lower curve. In this problem, the outer radius is the distance from the x-axis to $$y = 1$$, which is simply 1, and the inner radius is the distance from the x-axis to $$y = \ln x. So $$the volume integral becomes $$V = \int_{x=1}^{x=e} \pi \$$left(1^2$$ - (\ln x)^2$ight) \, dx$$ where $$e is $$the solution from step 1. Write the integral explicitly as $$V = \pi \$$int_1$$^{e} \left(1 - (\ln x)^2$ight) \, dx. $$This integral represents the volume of the solid generated by revolving the region R about the x-axis. To solve the integral, split it into two parts: $$\pi \$$int_1$$^{e} 1 \, dx - \pi \$$int_1$$^{e} (\ln x)^2$ \, dx. $$The first integral is straightforward, and the second integral can be solved using integration by parts or a known formula for $$\int (\ln x)^n \, dx. $$After evaluating both integrals, combine the results to express the volume.

Finding volume
Let R be the "triangular" region in the first quadrant that is bounded above by the line y = 1, below by the curve y = ln x, and on the left by the line x = 1.
Find the volume of the solid generated by revolving R about
a. the x-axis.

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Region Bounded by Curves

Volume of Solids of Revolution

Disk/Washer Method