Answer:
The volume of the solid formed by rotating the region bounded by y = x² and y = 1 about the line y = 7 can be found using the method of cylindrical shells and the integral V = 2π ∫ (7-y)(y) dy, evaluated from y = 1 to y = x² for x = -1 to x = 1.
Step-by-step explanation:
To find the volume of the solid obtained by rotating the region bounded by y = x², y = 1 about the line y = 7, we need to use the method of cylindrical shells.
The volume V of a solid of revolution generated by revolving a region about a horizontal line can be found by using the following integral:
V = 2π ∫ (radius)(height) dx
In this case, the radius is the distance from the point on the curve to the axis of rotation, which is 7 - y. The height is determined by the curve y = x², so the height is x².
The region of interest is bounded below by y = 1, so we rotate the region between y = 1 and y = x² from x = -1 to x = 1 (since x² = 1 gives x = ±1). The integral for the volume becomes:
V = 2π ∫-1∫1 (7-x²)(x²) dx
Upon evaluating the integral, we would find the volume of the solid.
This integral requires the use of integration techniques to be properly evaluated.