Final answer:
To find the volume of the solid obtained by rotating the region bounded by the curves y=x², x=5, and y=0 about the x-axis, we can use the method of cylindrical shells. The volume is approximately 1250π/4 or 982.06 cubic units.
Step-by-step explanation:
To find the volume of the solid obtained by rotating the region bounded by the curves y=x², x=5, and y=0 about the x-axis, we can use the method of cylindrical shells. The volume is given by the formula V = 2π ∫[a, b] x * (f(x) - g(x)) dx, where a and b are the x-coordinates of the intersection points of the curves, f(x) is the upper curve (y = x²), and g(x) is the lower curve (y = 0). In this case, a = 0 and b = 5, so the volume becomes V = 2π ∫[0, 5] x * (x² - 0) dx.
Simplifying the integral, we have V = 2π ∫[0, 5] x³ dx. Integrating x³ with respect to x, we get V = 2π [(1/4)x⁴] from 0 to 5, which simplifies to V = 2π [(1/4)(5⁴) - (1/4)(0⁴)]. Evaluating this expression, we find V = 2π (625/4), which is approximately equal to 1250π/4 or 982.06 cubic units.