Question

Volume of a sphere Let R be the region bounded by the upper half of the circle x²+y² = r² and the x-axis. A sphere of radius r is obtained by revolving R about the x-axis.

a. Use the shell method to verify that the volume of a sphere of radius r is 4/3 πr³.

Accepted Answer

Identify the region R bounded by the upper half of the circle $$x^{2}$$ + $$y^{2}$$ = $$r^{2}$$ and the x-axis. This means $$y$$ ranges from 0 to $$\sqrt{r$$^{2}$$ - x$$^{2}$$}$$ for $$x in$$ $$[-r, r]. $$Since we are revolving the region R about the x-axis, set up the shell method using vertical shells. The shells will be formed by slicing the region horizontally at a height $$y$$, where $$y$$ ranges from 0 to $$r. $$For the shell method, the radius of a shell is the distance from the x-axis to the shell, which is simply $$y. $$The height of the shell is the length of the horizontal segment inside the circle at height $$y$$, which can be found by solving for $$x$$: $$x = \pm \sqrt{r$$^{2}$$ - y$$^{2}$$}. So $$the height is $$2 \sqrt{r$$^{2}$$ - y$$^{2}$$}. $$The volume of each cylindrical shell is given by the formula $$dV = 2 \pi 	imes (	ext{radius}) 	imes (	ext{height}) 	imes (	ext{thickness}). $$Here, thickness is $$dy$$, so $$dV = 2 \pi y \cdot 2 \sqrt{r$$^{2}$$ - y$$^{2}$$} \ dy = 4 \pi y \sqrt{r$$^{2}$$ - y$$^{2}$$} \ dy. $$Integrate the volume of the shells from $$y=0 to$$ $$y=r to $$find the total volume: $$V = \int_{0}^{r} 4 \pi y \sqrt{r$$^{2}$$ - y$$^{2}$$} \ dy. $$Evaluate this integral (using substitution if needed) to verify that the volume equals $$\frac{4}{3} \pi r$$^{3}$$.

Volume of a sphere Let R be the region bounded by the upper half of the circle x²+y² = r² and the x-axis. A sphere of radius r is obtained by revolving R about the x-axis.

a. Use the shell method to verify that the volume of a sphere of radius r is 4/3 πr³.

Verified video answer for a similar problem:

Key Concepts

Shell Method for Volume

Equation of a Circle and Region Definition

Volume Formula of a Sphere

Volume of a sphere Let R be the region bounded by the upper half of the circle x²+y² = r² and the x-axis. A sphere of radius r is obtained by revolving R about the x-axis.a. Use the shell method to verify that the volume of a sphere of radius r is 4/3 πr³.

Verified video answer for a similar problem:

Key Concepts

Shell Method for Volume

Equation of a Circle and Region Definition

Volume Formula of a Sphere

Volume of a sphere Let R be the region bounded by the upper half of the circle x²+y² = r² and the x-axis. A sphere of radius r is obtained by revolving R about the x-axis.

a. Use the shell method to verify that the volume of a sphere of radius r is 4/3 πr³.