Let . A student claims that equals the limit of any Riemann sum of on as the…
MATHEMATICS · Single correct · AP · Calculus AB · Integration and Accumulation of Change · Integration and Accumulation of Change
Problem
Let g(x)={1,2,0≤x<11≤x≤2. A student claims that ∫02g(x)dx equals the limit of any Riemann sum of g on [0,2] as the partition is refined. Which statement correctly evaluates this claim?
(a) The claim is false because the jump discontinuity at x=1 prevents the upper and lower sums from converging to the same value.
(b) The claim is true because the value of g is bounded on [0,2].
(c) The claim is true because g has only one point of discontinuity, which does not affect the limit of Riemann sums.
(d) The claim is false because the limit of the Riemann sums can vary with the choice of sample points when a function has a jump discontinuity.