Fundamental Theorem of Calculus II

The second part of the Fundamental Theorem of Calculus connects antiderivatives with definite integrals. It finds the exact total accumulation of a function's rate over a specific interval.

If a function is continuous on a closed interval, the definite integral equals the antiderivative's value at the upper limit minus its value at the lower limit.

This avoids Riemann sum approximations, which estimate the area using many narrow rectangles under the curve.

The antiderivative chosen does not affect the final result. For instance, G of x may differ from another antiderivative by a constant C, but C cancels when subtracting values at the interval’s endpoints.

A common application is finding an object’s net displacement using its velocity function.

The net displacement for a given time interval, starting from 1 second to 4 seconds, is the area under the velocity-time graph.

To calculate this, first find the antiderivative of the velocity function—this gives the position function. Then, using the theorem, evaluate the position at the interval’s endpoints. Subtract the value of the position function at 1 second from its value at 4 seconds to get the net displacement.