Final answer:
The student requested assistance in finding the volume of a solid generated by revolving the area between y=10x√ and y=10x³ about the y-axis using the shell method. The volume calculation involves integrating the difference between the two functions over the interval where they intersect, from x = 0 to x = 1.
Step-by-step explanation:
The student has asked about finding the volume of a solid generated by revolving a region bounded by the curves y=10\( x \) and y=10x^3 about the y-axis using the shell method. The curves intersect at a point which can be found by setting the two equations equal to each other and solving for x, 10\( x \) = 10x^3. This yields the points of intersection as x = 0 and x = 1, using the fact that for x > 0, \( x \) = x^2. The volume V of the solid can be determined by the formula V = 2\( \pi \)\int_{a}^{b} x(f(x) - g(x)) dx, where f(x) and g(x) are our functions and [a, b] is the interval over which they are revolving. Therefore, the resulting volume to be calculated using the shell method is V = 2\( \pi \)\int_{0}^{1} x(10x^3 - 10\( x \)) dx. This integral would then be solved to find the exact volume.