The function f(x) = x - 2 is negative on the interval [0, 2] and positive on [2, 4]. What does the definite integral from 0 to 4 of f(x) dx represent?
AThe total area between the curve and the x-axis on [0, 4]
BThe signed area: the area above the x-axis minus the area below it
CTwice the area under the curve, because the interval has length 4
DZero, because the function crosses the x-axis
The definite integral computes signed area. On [0, 2] the function is below the x-axis, contributing negative area; on [2, 4] it is above, contributing positive area. The integral gives the net result. If the two pieces happen to be equal, the integral is 0 — but that is a coincidence of this function, not a general rule about functions that cross the axis.
Question 2 True / False
The definite integral of a continuous function f from a to b usually equals the area enclosed between the curve y = f(x) and the x-axis on [a, b].
TTrue
FFalse
Answer: False
The definite integral computes signed area: regions where f(x) < 0 contribute negative values. To find the geometric area (always non-negative), you must integrate the absolute value of f, or split the integral at the zeros and take the absolute value of each piece.
Question 3 Short Answer
What is the conceptual difference between a definite integral and an indefinite integral?
Think about your answer, then reveal below.
Model answer: A definite integral has specific bounds a and b and evaluates to a number (the signed area). An indefinite integral has no bounds and evaluates to a family of functions (the antiderivatives), written with a +C.
This distinction is frequently blurred, but it is fundamental. The definite integral is a completed calculation that produces a scalar. The indefinite integral is a question about which functions have a given derivative — its output is a function (or family of functions), not a number.