Fundamental Theorem of Calculus
Connect signed accumulated area A(x)=integral from a to x of (t^2-c) dt with A'(x)=x^2-c. Use the exact cubic antiderivative.
Free theory articleConnect signed accumulated area A(x)=integral from a to x of (t^2-c) dt with A'(x)=x^2-c. Use the exact cubic antiderivative.
Concept & interpretation
The Fundamental Theorem of Calculus bridges the concepts of differentiation and integration. It shows that accumulating area under a curve f(t) creates a new function A(x), whose derivative is the original function: A'(x) = f(x).
Interactive Experiment
Use the Fundamental Theorem of Calculus module to visually confirm this relationship:
- Set Lower bound a to 0 to fix the start of our accumulation.
- Set Offset c to 0 so the integrand is f(t) = t2.
- Set Difference step h to 0.5 to see the secant approximation.
- Set Upper position x to 1. Observe the shaded area representing 1/3.
- Notice that the Exact Derivative is exactly 1, while the Central Secant gives 13/12 (≈1.083333).
- Change Offset c to 1 and Lower bound a to -1. If you set Difference step h to 0.25, the exact area becomes -4/3 and the exact derivative becomes 0, with the central secant reflecting a small error of 1/48.
- Set Lower bound a and Upper position x both to 1 with Offset c = 0. Signed accumulation is 0 but Exact derivative remains 1.
- Set Lower bound a = 0, Upper position x = 1 and Offset c = 2. Read signed accumulation −5/3 and derivative −1; the graphs retain the signs.
- Clear Upper position x to see the error replace the dependent results. Enter 1 again to recover the calculation. Drag a plot and choose Reset view; all inputs stay unchanged.