What does the definite integral notation ∫_0^4 f(x) dx represent?

• A. The indefinite integral of f
• B. The derivative of f at x = 4
• C. The definite integral of f from 0 to 4
• D. The limit of f as x approaches 4

Answer: (C)

The definite integral from 0 to 4 of f(x) dx represents the total accumulation of f(x) as x runs from 0 to 4. The limits specify the interval, and dx indicates the variable of integration. This quantity is the net area between the curve y = f(x) and the x-axis over that interval (positive areas where f is above the axis and negative where it’s below). If f has an antiderivative F, the definite integral equals F(4) − F(0). This distinguishes it from an indefinite integral, which is an antiderivative with an added constant, and from a derivative or a limit of f, which are different operations. So the notation precisely means the definite integral of f from 0 to 4.