How do you find the volume of the resulting solid by any method of #x^2+(y-1)^2=1 # about the x-axis?

Question

How do you find the volume of the resulting solid by any method of #x^2+(y-1)^2=1 #        about the x-axis?

Samantha Adkison · Accepted Answer

Simplest must be to use Pappus' 2nd Theorem

Wiki Verbatim: "The second theorem states that the volume V of a solid of revolution generated by rotating a plane figure F about an external axis is equal to the product of the area A of F and the distance d traveled by its geometric centroid." Wiki

Christopher Allers · Answer

undefined To find the volume of the resulting solid formed by rotating the curve $x^2+(y-1)^2=1$ about the x-axis, you can use the method of cylindrical shells or the method of washers. Both methods involve integrating the cross-sectional area along the x-axis.

Using the method of cylindrical shells, you would integrate $2\pi x \cdot h(x) dx$, where $h(x)$ represents the height of the shell at each point along the x-axis.

Using the method of washers, you would integrate $\pi [R(x)^2 - r(x)^2] dx$, where $R(x)$ is the outer radius and $r(x)$ is the inner radius of each washer at a given x-coordinate.

For this particular curve, $x^2+(y-1)^2=1$, you would first solve for $y$ to express $y$ in terms of $x$, and then determine the appropriate limits of integration based on the intersection points of the curve with the x-axis. Once you have the expression for $y(x)$, you can proceed with the chosen method of integration to find the volume of the resulting solid.

Emma Aldridge · Answer

undefined To find the volume of the solid formed by rotating the region defined by $x^2 + (y - 1)^2 = 1$ about the x-axis, we can use the method of cylindrical shells.

The given equation represents a circle centered at $(0, 1)$ with radius 1.

The general formula for finding the volume using cylindrical shells is:

$$ V = 2\pi \int_a^b x \cdot f(x) \, dx $$

where $ f(x) $ represents the height of the shell at position $ x $, and $ a $ and $ b $ are the bounds of integration.

In this case, we can solve the equation $ x^2 + (y - 1)^2 = 1 $ for $ y $ to get $ y = 1 \pm \sqrt{1 - x^2} $. Since we're revolving around the x-axis, the height of the shell is $ f(x) = 2\sqrt{1 - x^2} $.

The bounds of integration are the x-values where the circle intersects the x-axis, which are $ x = -1 $ and $ x = 1 $.

So, the volume $ V $ is given by:

$$ V = 2\pi \int_{-1}^{1} x \cdot 2\sqrt{1 - x^2} \, dx $$

This integral can be evaluated using various integration techniques such as trigonometric substitution or by using tables of integrals.

After calculating the integral, you'll obtain the volume of the solid formed by rotating the given region about the x-axis.