q 6 a suppose a region r that is above the the x axis has an area give

Question

Q.6)
(a) Suppose a region R, that is above the the x-axis, has an area given by A. If that region
is rotated about the x-axis, the volume of revolution is given by V₁. If the same region is
rotated about the line y = −k (where k > 0), the volume of revolution is given by V₂. Derive
an expression for V₂ in terms of V₁, k and A. You should include a suitable diagram in your
derivation. Comment on how you would verify that the result you obtained is correct.
(b) Consider a function of two variables f(x, y) which has partial derivatives of all order. For all
values of t, u and u, the function satisfies f(tu, tv) = t²f(u, v). Derive a simplified expression
forin terms of f(x, y).
(c) Consider the function f(x) = x and assume that f(0) = 1. Using an appropriate established
MacLaurin series, and term by term integration, derive an infinite series expression for the
integral
cl