a consider the shaded area in the diagram below i by using integrals f

Question

(a) Consider the shaded area in the diagram below. i) By using integrals, find an expression for the shaded area. ii) Hence find the exact geometric area. (b) The volume of the solid formed by rotating the graph of y = f(x) about the x-axis between x = a an x = b is given by ii) Using the formula above, calculate the volume of the solid found in part(i). iii) Verify your result to part (ii) using a well-known formula. \int_{a}^{b} \pi[f(x)]^{2} d x \text { i) Describe the solid formed when the curve } y=\sqrt{64-x^{2}} \text { is rotated about } the x-axis.