The volume of the solid formed by revolving the region bounded by the graph of y = (x-3)² and the coordinate axes about the x-axis is given by what integral?
To find the volume of the solid formed by revolving the region bounded by the graph of y = (x-3)² and the coordinate axes about the x-axis, we can use the method of cylindrical shells.
The height of each shell will be the y-value of the function y = (x-3)², and the radius of each shell will be the x-value of the corresponding point on the graph.
To set up the integral for the volume, we will integrate over the range of x-values that corresponds to the region bounded by the graph, which in this case is from x = 0 to x = 3.
The volume of the solid can be calculated using the following integral:
V = ∫[0 to 3] 2πx * (x-3)² dx
To evaluate this integral, we will expand the squared term and distribute the coefficients:
V = ∫[0 to 3] 2πx * (x² - 6x + 9) dx
Next, we can distribute the x term:
V = ∫[0 to 3] 2πx³ - 12πx² + 18πx dx
To calculate this integral, we can use the power rule for integration. Integrating each term separately, we get:
V = 2π * [x⁴/4] - 12π * [x³/3] + 18π * [x²/2] evaluated from x = 0 to x = 3
Simplifying each term and plugging in the limits of integration, we get:
V = 2π * (81/4 - 0) - 12π * (27/3 - 0) + 18π * (9/2 - 0)
Further simplifying, we have:
V = 2π * 81/4 - 12π * 27/3 + 18π * 9/2
V = (162π - 324π + 162π)/4
V = 162π/4
V = 81π/2
Therefore, the volume of the solid formed by revolving the region bounded by the graph of y = (x-3)² and the coordinate axes about the x-axis is given by the integral V = 81π/2.
To find the volume of the solid formed by revolving the region bounded by the graph of y = (x-3)² and the coordinate axes about the x-axis, we can use the method of cylindrical shells and express the volume as an integral.
Here's how you can determine the integral:
1. First, let's identify the range of x-values for which the region is bounded. The graph of y = (x-3)² is a parabola with its vertex at (3, 0), so the region is bounded by x = 3.
2. Next, consider an infinitesimally thin vertical strip within this region, centered at x, with thickness dx. When this strip is revolved around the x-axis, it forms a cylindrical shell with height (x-3)² and thickness dx.
3. The circumference of the shell is equal to the circumference of the circular shape it forms, which is 2π times the radius. The radius is simply x.
4. The volume of this cylindrical shell can be calculated as its height times the circumference, which gives us 2πx(x-3)²dx.
5. To find the total volume of the solid, we need to integrate the cylinder shells over the range of x = 0 to x = 3.
Therefore, the integral that represents the volume of the solid is:
∫ from x = 0 to x = 3 of 2πx(x-3)²dx.
You can evaluate this integral using integration techniques such as u-substitution or expand it using algebra before integrating.