Find the volume of the solid generated by revolving the region bounded by the curves: y=2 Sqrt(x), y=4, x=0 about the x-axis
To find the volume of the solid generated by revolving the region bounded by the curves around the x-axis, we can use the method of integral calculus.
Step 1: Sketch the region bounded by the curves. The region is bound by the curve y = 2√x, the line y = 4, and the x-axis. Make sure to visualize the region in the positive x-axis region.
Step 2: Determine the limits of integration. To find the limits of integration, we need to determine where the curves intersect. We can set 2√x = 4 and solve for x to find the x-value where the two curves intersect.
2√x = 4
√x = 2
x = 4
Therefore, the limits of integration will be from x = 0 (starting point) to x = 4 (ending point).
Step 3: Set up the integral. We will use the disk method to calculate the volume. The formula for the volume using the disk method is:
V = ∫[a to b] π * (R^2 - r^2) dx
Where R is the outer radius and r is the inner radius.
In this case, the outer radius (R) is the value of y = 4, and the inner radius (r) is the value of y = 2√x.
So the volume formula becomes:
V = ∫[0 to 4] π * ((4)^2 - (2√x)^2) dx
Step 4: Simplify the equation. Applying the power rule and simplifying the equation gives us:
V = ∫[0 to 4] π * (16 - 4x) dx
Step 5: Evaluate the integral. Integrating the equation gives us:
V = π * [16x - 2x^2] |[0 to 4]
V = π * [(16 * 4 - 2 * 4^2) - (16 * 0 - 2 * 0^2)]
V = π * (64 - 32)
V = 32π
Therefore, the volume of the solid generated by revolving the region bounded by the curves y = 2√x, y = 4, and x = 0 about the x-axis is 32π cubic units.