Final answer:
To find the volume of the solid generated by revolving the region bounded by the line y=0 and the curve y=sqrt(4-x²) about the x-axis, we can use the method of cylindrical shells.
Step-by-step explanation:
To find the volume of the solid generated by revolving the region bounded by the line y=0 and the curve y=sqrt(4-x²) about the x-axis, we can use the method of cylindrical shells. The formula for the volume of a solid generated by revolving a region about the x-axis is V = 2π ∫[a,b] (x)(f(x)) dx, where a and b are the limits of integration and f(x) is the height of the curve at x. In this case, a = -2 and b = 2. Substituting the values into the formula, we have V = 2π ∫[-2,2] (x)(sqrt(4-x²)) dx.
Now, we can simplify the integral using a u-substitution. Let u = 4 - x². Then, du/dx = -2x, or dx = -(1/2x)du. Substitute these values into the integral to get V = -π ∫[-2,2] (x)(sqrt(u)) (1/2x) du.
Simplifying further, we have V = -π ∫[-2,2] sqrt(u) du. Integrate this with respect to u to get V = -π [(2/3)u^(3/2)]|[-2,2]. Evaluating the integral, we have V = -π [(2/3)(4-(-4))^(3/2) - (2/3)(4-(-4))^(3/2)]. Finally, simplify the expression to get V = (16/3)π