91Ó°ÊÓ

When you rotate a two-dimensional shape about an axis, you create a three-dimensional solid. The concept known as "volume of revolution" helps in finding the volume of such solids. In the exercise, we are looking at the region between two curves. Upon revolving this region around the specified axis, we can determine the volume of the resulting solid.

For this particular problem, we use the method of cylindrical shells. This involves integrating with respect to one variable, allowing us to "stack" cylinder-shaped shells to build up the volume of the solid. The volume is ultimately determined by measuring how much space these rotated shells take up as they combine to create the final shape.

Using the cylindrical shell method is particularly useful when rotating about the y-axis, as it simplifies the calculations by creating cylindrical shells that are easy to integrate over when stacked together.

Before we can find the volume of the revolution, we need to understand where the curves intersect. The intersection points mark the limits for the integration, setting the bounds for which the region is enclosed.

In the given exercise, to find the intersection points, we equate the two functions:

• The equation for the first curve is: \( y = 4x - x^2 \)
• The second curve is: \( y = x \)

Setting these equal, we solve:\[ 4x - x^2 = x \]Rearranging gives\[ x^2 - 3x = 0 \].Factoring this equation results in the solutions \( x = 0 \) and \( x = 3 \).

These points of intersection define the limits \((0, 3)\) over which we calculate the volume by integrating the area between these curves.

The concept of a definite integral is fundamental when calculating volumes of revolution. A definite integral allows us to find the exact area under a curve, between two specific points. In this exercise, it helps in determining the "integral" part of the cylindrical shell method formula.

The given integral calculation is:\[ V = 2\pi \, \int_{0}^{3} (3x^2 - x^3) \, dx \]
By performing this integration, we calculate:\[\int_{0}^{3} (3x^2 - x^3) \, dx = [x^3 - \frac{x^4}{4}]_{0}^{3} \]
We find this definite integral by evaluating the antiderivative at the upper limit and subtracting the value of the antiderivative at the lower limit.

This exact value gives the "integral shell," or space each shell occupies, and establishes the groundwork to further complete the total volume calculation.

The action of rotating a region about the y-axis creates a solid of revolution. It can sometimes seem complex, but the visualization helps in understanding how the shells build up the entire solid.

Consider a typical point on your curve. When this point, along with others in its vicinity, is rotated around the y-axis, you generate a series of concentric cylindrical shells. Each shell is a cross-sectional part of this generated solid.

The height of each shell, which varies across the x-values due to the intersection of those curves, dictates the portion of volume contributed by each shell. The shell method formula incorporates all these measurements:\[ V = 2\pi \, \int_{a}^{b} x \, (f(x) - g(x)) \, dx \]Collectively integrating these shells from the intersection points results in the full shape, and thus, the overall volume.