Final answer:
The volume of the solid formed by revolving the graph of y = -x² around the x-axis is found using disk integration. The radius of each disk is represented by the function y, leading to the integral ∫_a^b πx⁴ dx, which is evaluated between the x-intercepts of the parabola.
Step-by-step explanation:
When calculating the volume of a solid formed by revolving the region bounded by the curve y = -x² about the x-axis, we use the method of disk integration to set up the definite integral. This method slices the solid into thin disks perpendicular to the x-axis.
To find the volume, V, of the solid, we express the radius of each disk as a function of x, which in this case is the given y-function. Thus, the area of each disk is πr² = π(-x²)² = πx⁴. We then integrate this area from the leftmost to the rightmost point of the region in question, usually the x-intercepts of the function.
The definite integral representing the volume is given by:
V = ∫_a^b π(-x²)² dx = ∫_a^b πx⁴ dx
Where a and b are the x-intercepts of the parabola y = -x². To evaluate this integral, we apply the power rule for integration, integrating x⁴ term by term and then evaluate the resulting antiderivative at b and a, subtracting the latter from the former.