Shell Method HW

Front

What is the Shell Method in calculus?

Back

The Shell Method is a technique for finding the volume of a solid of revolution. It involves integrating the lateral surface area of cylindrical shells formed by revolving a region around an axis.

Front

How do you set up the integral for the Shell Method when revolving around the y-axis?

Back

To set up the integral for the Shell Method when revolving around the y-axis, use the formula: \( V = 2\pi \int_{a}^{b} (radius)(height) \, dx \), where 'radius' is the distance from the y-axis and 'height' is the function value.

Front

What is the formula for the volume of a solid of revolution using the Shell Method?

Back

The volume \( V \) is given by: \( V = 2\pi \int_{a}^{b} (radius)(height) \, dx \) for rotation around the y-axis, and \( V = 2\pi \int_{c}^{d} (radius)(height) \, dy \) for rotation around the x-axis.

Front

What is the radius in the Shell Method when revolving around the y-axis?

Back

The radius is the distance from the axis of rotation (y-axis) to the shell, which is typically the x-coordinate of the function being revolved.

Front

What is the height in the Shell Method when revolving around the y-axis?

Back

The height is the value of the function being revolved, which represents the vertical distance of the shell.

Front

How do you find the volume of the solid generated by revolving the region bounded by \( y = 2\sqrt{x} \) about the y-axis?

Back

Set up the integral using the Shell Method: \( V = 2\pi \int_{0}^{4} (x)(2\sqrt{x}) \, dx \) and evaluate.

Front

What is the significance of the limits of integration in the Shell Method?

Back

The limits of integration define the interval over which the region is being revolved, determining the bounds of the volume calculation.

Front

How do you find the volume of the solid obtained by rotating the area between the graphs of \( y = x^2 \) and \( x = 2y \) around the y-axis?

Back

Use the Shell Method: \( V = 2\pi \int_{0}^{2} (y)(2y) \, dy \) and evaluate the integral.

Front

What is the relationship between the Shell Method and the Washer Method?

Back

Both methods are used to find volumes of solids of revolution, but the Shell Method uses cylindrical shells while the Washer Method uses washers (disks) to calculate volume.

Front

What is the volume of the solid generated by revolving the region bounded by \( y = 2 - x \), \( y = 0 \), and \( x = 0 \) around the x-axis?

Back

Set up the integral using the Washer Method: \( V = \pi \int_{0}^{2} ((2-x)^2 - 0^2) \, dx \) and evaluate.