The antiderivative of a function provides a geometric interpretation related to the accumulation of quantities. When we find the antiderivative F(x) of a function f(x), we are essentially finding a function whose rate of change, or slope, is given by f(x). Geometrically, if you graph f(x), the antiderivative F(x) will give a graph whose slope at any point x is equal to the value of f(x) at that point. In other words, F(x) accumulates the values of f(x) as x increases, and the rate at which it accumulates is precisely f(x). This is particularly insightful in physics, where the antiderivative of velocity gives displacement, accumulating the velocity to give total change in position.

Yes, the integral of a function can be negative, and it signifies that the function lies below the x-axis over the interval of integration. When we calculate the definite integral, we are finding the net area between the function and the x-axis. If the function is above the x-axis, the integral is positive, representing a positive area. If the function is below the x-axis, the integral is negative, indicating that the function traverses below the axis. In physical terms, a negative integral might represent a loss, such as a negative work done or a deficit in an economic model, depending on the context of the problem.

Integration is widely used in physics to describe various aspects of motion, particularly when dealing with variables like velocity and acceleration. For instance, if acceleration (a function of time) is known, integration can be used to find the velocity function by finding the antiderivative of the acceleration. Similarly, if the velocity function is known, the position or displacement of an object can be determined by integrating the velocity function. This is because the integral accumulates the rate of change provided by these functions, offering a total change (in velocity or position) over a specific time interval. This principle is foundational in kinematics, enabling physicists to model and predict the motion of objects under various conditions.

Integration is fundamentally connected to finding the area under a curve due to its ability to sum up infinitesimally small areas. When we integrate a function from a to b, we are essentially adding up all the tiny areas under the curve y = f(x) from x = a to x = b. The integral takes into account the height of the function at every point across the interval [a, b] and accumulates these values, providing the net area under the curve. This concept is particularly useful in physics and engineering, where it is often necessary to calculate quantities related to areas, such as work done or electric charge.

The constant of integration, often denoted as C, is added when finding the antiderivative due to the fundamental theorem of calculus. When we differentiate a constant, the result is zero. Therefore, when we find the antiderivative, we must account for any possible constant that might have been present in the original function before differentiation. Including C ensures that all possible original functions are represented. For example, the derivative of both x² and x² + 5 is 2x. So, when we find the antiderivative of 2x, we express it as x² + C to account for any constant that might have been in the original function.