Accumulation and the Meaning of the Definite Integral
Water flows into a reservoir at a rate r(t) measured in liters per minute, where r(t) is positive but decreasing over a 10-minute period. A student computes the left Riemann sum L5 with 5 equal subintervals and the right Riemann sum R5 with the same subintervals, and finds L5=820 L and R5=760 L. The student is unsure which is the better estimate of the total volume added and whether the definite integral equals either sum. The teacher asks the student to decide whether the true accumulated volume lies between the two sums, to explain why, and to connect the answer to the interpretation of the definite integral as a limit of Riemann sums rather than as a particular sum.
1. Given r is decreasing, which of L5 and R5 overestimates and which underestimates the true accumulated volume? Justify with the geometry of rectangles.
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For a decreasing function, on each subinterval the left endpoint gives the largest value, so the left-endpoint rectangle sits above the curve: L5 overestimates. The right endpoint gives the smallest value, so R5's rectangles sit below the curve: R5 underestimates. Therefore the true accumulated volume V = integral from 0 to 10 of r(t) dt satisfies R5 <= V <= L5, i.e., 760 <= V <= 820. The monotonicity of r is what guarantees the ordering; if r were not monotone, neither sum would be a guaranteed bound.
2. Explain why the definite integral is not equal to L5 or R5 in general, and what limit defines it.
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The definite integral is defined as the limit of Riemann sums as the partition norm goes to 0: integral from 0 to 10 of r(t) dt = lim_{n->infinity} sum_{i=1}^{n} r(t_i*) Delta t, for any choice of sample points t_i* in each subinterval, provided the limit exists. L5 and R5 are single sums with a fixed partition (n=5); they are approximations, not the limit. As n increases the gap between left and right sums shrinks, and both converge to the same value, the integral. Since r is continuous, the integral exists and equals that common limit; L5 and R5 bracket it but are not equal to it.
3. The student claims 'since the function is decreasing, the integral equals the average of L5 and R5.' Evaluate this claim.
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The claim is false in general. The average (820+760)/2=790 is a plausible midpoint estimate, but there is no theorem that the true integral equals the average of the left and right sums. The trapezoidal rule T5 = (L5+R5)/2 = 790 is a specific approximation, and for a function with curvature it has its own error; it is not exact unless r is linear on each subinterval. The correct statement is only the bracketing 760 <= V <= 820. Confusing an approximation formula with an identity is the error. The integral is the limit of sums, not the mean of two particular sums.