Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis.
y=6x^2, x=1, y=0, about the x-axis
To find the volume of the solid obtained by rotating the region bounded by the curves about the x-axis, we can use the method of disk integration.
First, let's visualize the region bounded by the curves. The curve y = 6x^2 is a parabolic curve that opens upward. The line x = 1 is a vertical line passing through the point (1,0). The region bounded by these curves is a segment of the area under the parabola between x = 0 and x = 1.
To calculate the volume of the solid, we will integrate the area of the circular cross-sections obtained by rotating the region about the x-axis.
Now, let's find the equation of the parabolic curve in terms of y. Solving y = 6x^2 for x, we get x = √(y/6). By using this equation, we can determine the limits of integration.
Since we are rotating the region about the x-axis, the radius of each circular cross-section will be given by the value of x. Thus, the radius of each cross-section is √(y/6).
To find the limits of integration for y, we need to find the value of y when x = 1. Plugging x = 1 into the equation y = 6x^2, we get y = 6.
Therefore, the integral for the volume is given by:
V = ∫[0 to 6] π(√(y/6))^2 dy
Simplifying the integral:
V = π/6 * ∫[0 to 6] y dy
Evaluating the integral:
V = π/6 * [y^2/2] [from 0 to 6]
V = π/6 * [36/2 - 0]
V = π/6 * [18]
V = 3π cubic units
Hence, the volume of the solid obtained by rotating the region bounded by the curves y = 6x^2, x = 1, and y = 0 about the x-axis is 3π cubic units.