Question

35–38. Shell and washer methods Let R be the region bounded by the following curves. Use both the shell method and the washer method to find the volume of the solid generated when R is revolved about the indicated axis.

y = (x−2)³ −2,x=0, and y=25; about the y-axis

Accepted Answer

First, identify the region R bounded by the curves: $$y = (x - 2)^3$ - 2$$, the vertical line $$x = 0$$, and the horizontal line $$y = 25. $$Sketching or visualizing this region helps understand the limits of integration and the shape of the solid. For the shell method (revolving around the y-axis), express the radius and height of a typical cylindrical shell. The radius is the distance from the y-axis, which is $$x$$, and the height is the vertical distance between the curves, which can be written in terms of $$x as$$ $$y_{top} - y_{bottom}. $$Here, the height is $$25 - ((x - 2)^3$ - 2). $$Set up the integral for the shell method. The volume $$V is $$given by $$V = 2\pi \int_{a}^{b} (	ext{radius})(	ext{height}) \, dx. $$Determine the limits of integration $$a$$ and $$b by $$finding the $$x-$$values where the region is bounded, specifically between $$x=0$$ and the $$x-$$value where $$y=25$$ intersects $$y = (x-2)^3$ - 2. $$For the washer method (also revolving around the y-axis), express $$x as a $$function of $$y by $$solving $$y = (x - 2)^3$ - 2$$ for $$x. $$This gives $$x = 2 + \sqrt[3]{y + 2}. $$The outer radius is the distance from the y-axis to the outer curve, and the inner radius is the distance to the inner curve (here, $$x=0). $$Set up the integral for the washer method. The volume $$V is $$given by $$V = \pi \int_{c}^{d} \left[(	ext{outer radius})^2$ - (	ext{inner radius})^2$ight] dy. $$The limits $$c$$ and $$d$$ correspond to the $$y-$$values bounding the region, from $$y = -2 ($$when $$x=0) up to$$ $$y=25. $$Write the integral explicitly using these limits and the expressions for the radii.

35–38. Shell and washer methods Let R be the region bounded by the following curves. Use both the shell method and the washer method to find the volume of the solid generated when R is revolved about the indicated axis.

y = (x−2)³ −2,x=0, and y=25; about the y-axis

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Shell Method

Washer Method

Setting up the Region and Limits of Integration

35–38. Shell and washer methods Let R be the region bounded by the following curves. Use both the shell method and the washer method to find the volume of the solid generated when R is revolved about the indicated axis.y = (x−2)³ −2,x=0, and y=25; about the y-axis

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Shell Method

Washer Method

Setting up the Region and Limits of Integration

35–38. Shell and washer methods Let R be the region bounded by the following curves. Use both the shell method and the washer method to find the volume of the solid generated when R is revolved about the indicated axis.

y = (x−2)³ −2,x=0, and y=25; about the y-axis