Final answer:
To find the volume of the solid obtained by rotating the region, use the method of cylindrical shells by integrating 2πx^2 over the interval [0, 5]. The volume will be 250/3π cubic units.
Step-by-step explanation:
To find the volume V of the solid obtained by rotating the region bounded by the curves y = x, y = 0, x = 0, x = 5 about the x-axis, we can use the method of cylindrical shells. The volume of each cylindrical shell is given by 2πrhΔx, where r is the distance from the shell to the axis of rotation (which is the x-axis in this case), h is the height of the shell, and Δx is the thickness of the shell.
The height of each shell can be found by taking the difference between the upper and lower curves at a particular x-value. In this case, the upper curve is y = x and the lower curve is y = 0. So, the height of each shell is x - 0 = x.
The radius of each shell is simply the x-value. Thus, the volume of each shell is 2πx^2Δx. To find the total volume, we need to integrate this expression over the interval [0, 5]. So, the volume V is given by:
V = ∫[0,5] 2πx^2 dx
Integrating this expression gives:
V = [2/3πx^3] from x = 0 to x = 5
V = 2/3π(5^3 - 0^3)
V = 2/3π(125)
V = 250/3π cubic units