Final answer:
To find the volume of the solid obtained by rotating the region bounded by the curves y=7x^4 and y=7x about the x-axis, we can use the method of disks or washers. The volume can be found by integrating the expression πf(x)^2*dx over the interval [0, 1].
Step-by-step explanation:
To find the volume of the solid obtained by rotating the region bounded by the curves y=7x^4 and y=7x about the x-axis, we can use the method of disks or washers. First, let's sketch the region bounded by the curves. The curve y=7x^4 intersects the curve y=7x at the points (0, 0) and (1, 7).
When we rotate this region about the x-axis, we obtain a solid shape. To find the volume of this shape, we consider taking slices or disks perpendicular to the x-axis. These slices have radius f(x), where f(x) is the distance between the curve y=7x and the x-axis. The volume of each slice can be approximated as πf(x)^2*dx. By integrating this expression over the interval [0, 1], we can find the total volume of the solid.
By using integration, we can find that the volume V of the solid is:
V = ∫01 π(7x-7x^4)^2 dx