Which integral represents the area under the curve of a function from point a to b?

The integral that represents the area under the curve of a function from point a to b is defined as the definite integral of the function, which is mathematically expressed as ∫ f(x) dx from a to b. This integral calculates the accumulated area between the curve of the function f(x) and the x-axis, within the limits of a and b.

In the context of definite integrals, when you evaluate ∫ f(x) dx from a to b, you are essentially finding the net area between the curve and the x-axis. If the function is above the x-axis in the interval [a, b], this area is positive, while if it lies below the x-axis, the area is considered negative. The fundamental theorem of calculus confirms that integration and differentiation are inverse processes, reinforcing that the definite integral gives us the accumulation of the function's values over the specified interval.

The other expressions do not yield the area under the function itself:

• The integral of the derivative f'(x) over an interval would give you the change in the function's value, not the area under the curve.

• The second derivative f''(x) would represent the rate of change of the rate of change of the function, which again does not