# Second Fundamental Theorem of Calculus

- Author:
- Dr. Jack L. Jackson II

## Accumulation Function

In this activity we start with a function f(x) which is graphed in the left window. Enter the desired formula in the input box for f(x). For now set

*a*= 0,*x*= 0, and*C*= 0 via their sliders or input boxes. Now slide the slider for*x*slowly to the right. The shaded area is accumulating as we increase the value for*x*. Green areas accumulate positively and red areas accumulate negatively. The green area minus the red area is the value of the integral of*f*(*x*) over the interval [*a, x*]. This is demonstrated by the size of the vertical line segment in the window on the right. Note that this defines a new accumulation function*A*. Note that this function goes through the origin. Check the checkbox for*A*(*x*) in the right window. You will now see the entire accumulation function. On this blue accumulation function which is graphed in the right window,*y*-values are equal to the green area - red area from the window on the left. If we adjust the value of a, then the accumulation will now go through (*a*, 0) instead of the origin. If we adjust the value of*C*, then the entire accumulation function will be shifted vertically by*C*units. Note that this gives us a way to take any function we can integrate and use it to define a new related accumulation function. Actually, there are a whole family of these accumulation functions for different choices of*a*and*C*.## Second Fundamental Theorem of Calculus

(Fundamental Theorem of Calculus Part 2)
Click on the
This tells us that each of these accumulation functions are antiderivatives of the original function

*A*'(*x*) checkbox in the right window. This will graph the derivative of the accumulation function in red in the right window. How does*A*'(*x*) compare to the original*f*(*x*)? They are the same! This illustrates the**Second Fundamental Theorem of Calculus**For any function*f*which is continuous on the interval containing*a, x*, and all values between them:*f*. First integrating and then differentiating returns you back to the original function. In this sense, integration and differentiation are inverse processes.