Ive got another question today: Define $f : [−2, 2] →$ $\mathbb{R}$ by f(x) = $x^2 -4$. Let $Q = \{−2, 0, 1, 2\}$, and consider the corresponding Riemann sum $S_Q(f, ξ)$. What values can the Riemann sum $S_Q(f, ξ)$ get? Conclude that $\int_{-2}^{2} f(x) dx<0$
I can easily conclude that $\int_{-2}^{2} f(x) dx<0$ but cant really explain the first part of this question. Any help will be appreciated.
Well, for the upper Riemann sums. You can do this by finding the sup of the function evaluated at each subinterval of your partition Q that is the sup between f(-2),f(0) and then the sup at f(0),f(1) and so on.. After finding the sups of each subinterval you just multiply that sup with the distance of the corresponding subinterval and find the sum. Example: sup[f(-2),f(0)](-2-0) + sup[f(0),f(1)](0-1)+... Similar argument would be made for the lower Riemann sum but for the inf rather than sup.