Final answer:
The volume of the solid obtained by rotating the region bounded by y = 7 - 7x² and y = 0 about the x-axis is calculated using the method of disks and integration from the intersections of curves, which are x = -1 and x = 1.
Step-by-step explanation:
To find the volume of the solid obtained by rotating the region bounded by the curves y = 7 - 7x² and y = 0 about the x-axis, we can use the method of disks as it involves a rotation around the x-axis. First, we need to find the intersections of the two curves, which, in this case, are where the parabola meets the x-axis. Setting y = 0 in the parabolic equation gives us the roots x = ± 1.
The volume of the solid of revolution can be calculated using the formula for the volume of a disk: V = π ∫ [f(x)]² dx, from x_1 to x_2. Substituting the boundaries and the function into the formula, we get:V = π ∫ from -1 to 1 (7 - 7x²)² dxThis integral represents the sum of the volumes of infinitesimally thin disks with radius (7 - 7x²) centered on the x-axis from x = -1 to x = 1.