Final answer:
To determine the volume, we solve for the intersection points of y=7x² and y=7x√3, which are x = 0 and x = √3. Then, we use the disk method with respect to the y-axis and set up the integral to calculate the volume. The volume formula for a sphere is relevant as an analogy to remember the correct formula for volumes involving π and powers of a radius.
Step-by-step explanation:
To find the volume of the solid generated by revolving the region bounded by the curves y=7x² and y=7x√3 about the y-axis, we first identify the points of intersection by setting the two equations equal to each other: 7x² = 7x√3, which gives us the solutions x = 0 and x = √3. The sketch would show a parabola opening upwards for y=7x², and a straight line with a positive slope intersecting the parabola at the origin (0,0) and at ( √3, 7√3).
Using the disk method, we express the volume of the solid as an integral. Each disk has volume πr²h, where r is the radius, and h is the height. As we are revolving around the y-axis, we will take vertical slices, so the radius r = x and the height h is the difference between the two functions. The integral for the volume becomes V = π ∫_{0}^{7√3} (r_x² - r_x√3)^2 dy, where r_x = x(y) is the radius of the disk in terms of y.
After calculating the volume, we apply the correct volume formula; we know from basic geometry knowledge that for a sphere, the volume formula is (4/3)πr³, not to be confused with the formula for the surface area, which is 4πr².