Concept guide
Definite integral: computing ∫₀³(x²−4)dx as a limit of left sums and by the moment formula
Let f(x)=x²−4, a=0, b=3. (1) Using only the Riemann definition (the common limit of all valid tagged-partition sums as the mesh tends to zero) and the premise that f is integrable, find I(f,0,3) as the limit of the N-piece left sums L_N; (2) recompute it independently with the moment formula listed in this MKO and compare; (3) find the reversed integral I(f,3,0); (4) use N=3 and N=50 to show that a finite left sum is not the integral, and explain why the signed integral is not a nonnegative area.