Question

9-34. Shell method Let R be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when R is revolved about indicated axis.

y = x³−x⁸+1,y=1; about the y-axis

Accepted Answer

First, identify the region R bounded by the curves given: $$y = x$$^{3}$$ - x$$^{8}$$ + 1$$ and $$y = 1. $$Since the region is revolved about the y-axis, we will use the shell method with respect to $$x. $$Set up the shell radius and height. The radius of a typical shell is the distance from the y-axis, which is $$x. $$The height of the shell is the vertical distance between the curves, which is $$y - 1 = (x$$^{3}$$ - x$$^{8}$$ + 1) - 1 = x$$^{3}$$ - x$$^{8}$$. Determine the interval for x$ over which the region exists. Since the region is bounded by y = 1$ and y$$ = $$x^{3}$$ - $$x^{8} + 1$, find the values of x$ where these two curves intersect by solving x$$^{3}$$ - x$$^{8}$$ + 1 = 1$$, which simplifies to $$x^{3}$$ - $$x^{8} = 0$. Express the volume integral using the shell method formula: V = 2$$\pi \int_{a}^{b} (	ext{radius})(	ext{height}) \, dx = 2\pi \int_{a}^{b} x $$(x^{3}$$ - $$x^{8}$$) \, dx$$ where a$ and b$ are the intersection points found in the previous step. Simplify the integrand to x$$^{4}$$ - x$$^{9}$$ and set up the definite integral for volume: V = 2$$\pi \int_{a}^{b} $$(x^{4}$$ - $$x^{9}$$) \, dx$$. The final step would be to evaluate this integral to find the volume.

9-34. Shell method Let R be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when R is revolved about indicated axis.

y = x³−x⁸+1,y=1; about the y-axis

Verified video answer for a similar problem:

Key Concepts

Shell Method for Volume

Setting up the Integral with Given Curves

Revolution about the y-axis

9-34. Shell method Let R be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when R is revolved about indicated axis. y = x³−x⁸+1,y=1; about the y-axis

Verified video answer for a similar problem:

Key Concepts

Shell Method for Volume

Setting up the Integral with Given Curves

Revolution about the y-axis

9-34. Shell method Let R be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when R is revolved about indicated axis.

y = x³−x⁸+1,y=1; about the y-axis