To find the volume of the solid obtained by rotating the region bounded by the curves y = (x-3)^2, x = 0, and y = 4 about a specified line, we can use the disk method. Setting up the integral and solving it will give us the volume of the solid.
To find the volume of the solid obtained by rotating the region bounded by the curves y = (x-3)^2, x = 0, and y = 4 about a specified line, we can use the disk method.
The disk method involves considering cross-sections perpendicular to the axis of rotation and summing their volumes. First, let's find the points of intersection between the curves. Setting (x-3)^2 = 4, we find x = 1 and x = 5.
The axis of rotation is the line x = 3. Now, let's consider a cross-section at a specific x value. The radius of this disk is y and the height is dx. The volume of this disk can be expressed as πy^2 dx. Integrating this expression from x = 1 to x = 5 will give us the volume of the solid.
Setting up the integral:
Find the limits of integration: from 1 to 5.
Express y in terms of x: y = (x-3)^2.
Express dx in terms of x: dx = 1.
Write the integral: ∫(πy^2 dx) from 1 to 5.
Solving the integral will give us the volume of the solid obtained by rotating the region bounded by the given curves about the specified line.