A continuous rate function satisfies for and for

MATHEMATICS · Single correct · AP · Calculus AB · Applications of Integration · Applications of Integration

Problem

A continuous rate function gg satisfies g(t)>0g(t)>0 for 0≤t<40\le t<4 and g(t)<0g(t)<0 for 4<t≤94<t\le 9. A student states, “The total amount added to the related quantity between t=0t=0 and t=9t=9 equals ∫09g(t) dt.\displaystyle \int_{0}^{9} g(t)\,dt.” Which of the following best evaluates the student’s claim? (a) The claim is correct because the integral accumulates all contributions regardless of sign. (b) The claim is incorrect; since g(t)g(t) is negative after t=4t=4, the total amount added is ∫04g(t) dt\displaystyle \int_{0}^{4} g(t)\, dt. (c) The claim is incorrect because the integral gives the net change; negative values on (4,9](4,9] reduce the result, so the student should instead use ∫09∣g(t)∣ dt\displaystyle \int_{0}^{9} |g(t)|\,dt. (d) The claim is incorrect because the integral must be divided by the length of the interval to find the total amount added.

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