the following theorem. Theorem If f is integrable on [a, b], then b [ºr(x) dx f(x) dx = lim f(x₁)AX n→∞ where Ax = i = 1 (x² b-a n and x; = a + iAx. iven theorem to evaluate the definite integral. 9 [₁²³ 4x + 9) dx

Let f be a function that is defined within the interval [-1, 1] so that f(x) = the image attachment.  Show that f is continious in the entire interval of [-1, 1]

Consider the following theorem. Theorem If f is integrable on [a, b], then 9. n f(x) dx = lim S {(x;)Ax n> 00 i = 1 b where Ax = — а and x; * = a + ¡Ax, n Use the given theorem to evaluate the definite integral. (7x2 + 7x) dx -2

Let f(z) = u(x,y) + iv(x,y). If u (x,y) = 2x-x^3+3xy^2, determine v(x,y) such that f(z) is analytic.Use the Theorem below