Question

45–48. Shell and washer methods about other lines Use both the shell method and the washer method to find the volume of the solid that is generated when the region in the first quadrant bounded by y = x²,y=1, and x=0 is revolved about the following lines.

x = -1

Accepted Answer

First, identify the region bounded by the curves: $$y = x$$^{2}$$, y = 1$, and x = 0$ in the first quadrant. This region lies between x=0$ and x=1$ because when y=1$, x$$ = \sqrt{1} = 1$$. For the shell method about the line x = -1$, consider vertical slices parallel to the axis of rotation. The radius of a shell at position x$ is the distance from x$ to -1$, which is r = x - (-1) = x + 1$. The height of the shell is the vertical distance between y=1$ and y$$=$$x^{2}$$, so $$h = 1 - x$$^{2}$$. Set up the shell method integral for the volume V$ as:  
V = 2$$\pi \int_{0}^{1} (	ext{radius})(	ext{height}) \, dx = 2\pi \int_{0}^{1} (x + 1)(1 - $$x^{2}$$) \, dx$$. For the washer method about the line x = -1$, consider horizontal slices perpendicular to the axis of rotation. Express x$ in terms of y$: from y$$ = $$x^{2}$$, we get $$x = \sqrt{y}. $$The region extends from $$y=0 to$$ $$y=1. $$The outer radius $$R is $$the distance from $$x = -1 to $$the right boundary $$x=1$$, so $$R = 1 - (-1) = 2. $$The inner radius $$r is $$the distance from $$x = -1 to $$the curve $$x = \sqrt{y}$$, so $$r = \sqrt{y} - (-1) = \sqrt{y} + 1. $$Set up the washer method integral for the volume $$V as$$:  
$$V = \pi \int_{0}^{1} \$$left(R^{2}$$ - r$$^{2}$$ight) dy = \pi \int_{0}^{1} \$$left(2^{2}$$ - (\sqrt{y} + 1$$)^{2}$$ight) dy.$$

45–48. Shell and washer methods about other lines Use both the shell method and the washer method to find the volume of the solid that is generated when the region in the first quadrant bounded by y = x²,y=1, and x=0 is revolved about the following lines.

x = -1

Verified video answer for a similar problem:

Key Concepts

Shell Method for Volume

Washer Method for Volume

Adjusting for Axis of Rotation Not on Coordinate Axes

45–48. Shell and washer methods about other lines Use both the shell method and the washer method to find the volume of the solid that is generated when the region in the first quadrant bounded by y = x²,y=1, and x=0 is revolved about the following lines. x = -1

Verified video answer for a similar problem:

Key Concepts

Shell Method for Volume

Washer Method for Volume

Adjusting for Axis of Rotation Not on Coordinate Axes

45–48. Shell and washer methods about other lines Use both the shell method and the washer method to find the volume of the solid that is generated when the region in the first quadrant bounded by y = x²,y=1, and x=0 is revolved about the following lines.

x = -1