The result of a definite integral represents the signed area between a function’s graph and the x-axis over a specified interval. If the function is always above the x-axis within that interval, the resulting value is positive. However, if the function dips below the x-axis within the interval, the area below the x-axis contributes a negative value. For instance, integrating a function such as f(x) = x2 from 0 to 1 yields a positive result. Conversely, integrating f(x) = -x2 from 0 to 1 will yield a negative result.
Understanding that the computed value can be positive, negative, or zero is crucial in various applications. In physics, the integral of velocity with respect to time yields displacement; a negative displacement indicates movement in the opposite direction. In economics, the area under a marginal cost curve represents the total cost; a negative value would be nonsensical in this context, indicating a potential error in the model. The ability to correctly interpret the sign of the resulting value is key to meaningful analysis and problem-solving.
