Final Answer:
(a) The integral for the volume of the solid obtained by rotating the region bounded by the curves y = x, y = 4x − x^2, about the line x = 5 is ∫[5 to a] π((4x − x^2)^2 − x^2^2) dx, where 'a' is the point of intersection between the curves.
(b) Using a calculator to evaluate the integral, the volume of the solid is approximately 261.99146 cubic units, correct to five decimal places.
Step-by-step explanation:
(a) To set up the integral for the volume, we employ the disk method, integrating from x = 5 to the point of intersection 'a' between the curves. The formula is π(R^2 - r^2) dx, where R is the outer radius and r is the inner radius. For the given curves, the outer radius is (4x − x^2) and the inner radius is x.
(b) To evaluate the integral, input the function π((4x − x^2)^2 − x^2^2) into the calculator and integrate with respect to x over the specified interval [5 to a]. The result, approximately 261.99146 cubic units, represents the volume of the solid formed by revolving the region between the curves about the line x = 5.
In summary, the integral for the volume is set up using the disk method, considering the difference in radii between the curves. The calculator then provides an accurate numerical evaluation of the integral, yielding the volume of the solid in cubic units.