Second Fundamental Theorem of Calculus Calculator

Calculator Inputs

Integrand f(t)

Use t inside the integral.

Lower limit h(x)

Use x for moving limits.

Upper limit g(x)

Evaluate at x

Derivative step size

Simpson intervals

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Supported examples: sin(t), cos(t), sqrt(t), ln(t), exp(t), pi, and e. Use * for multiplication.

Example Data Table

Formula Used

For a single moving upper limit, the theorem is:

d/dx ∫[a, g(x)] f(t) dt = f(g(x))g'(x)

For two moving limits, this calculator uses the advanced form:

d/dx ∫[h(x), g(x)] f(t) dt = f(g(x))g'(x) - f(h(x))h'(x)

It estimates g'(x) and h'(x) with centered differences. It also estimates the integral with Simpson's rule for comparison.

How to Use This Calculator

1. Enter the integrand with t as the integration variable.
2. Enter lower and upper bounds with x as the outside variable.
3. Enter the point where the derivative should be evaluated.
4. Choose a small step size and an even Simpson interval count.
5. Press calculate to see the theorem result above the form.
6. Use CSV or PDF options to save the output.

Detailed Guide

Understanding the theorem

The second fundamental theorem of calculus links accumulation with rate of change. It says a variable upper limit turns an integral into a differentiable function. The derivative depends on the integrand at the moving limit. It also depends on the speed of that limit.

This idea looks simple. Yet it saves many steps in advanced calculus. You do not need to expand the whole integral first. You can evaluate the outside motion and inside function directly. When both limits move, the upper limit adds a term. The lower limit subtracts a term.

How the calculator works

This calculator follows that rule. It reads an integrand written in the variable t. It reads lower and upper limits written in the variable x. Then it evaluates each limit at your chosen x value. It estimates each limit derivative with a centered difference. It applies the theorem and shows every major component.

A numerical integration check is also included. The tool estimates the integral near your x value. It compares the slope of that accumulated area with the theorem result. Small differences can happen. They usually come from rounding, step size, or difficult functions.

Input tips

Use simple multiplication signs. Write 3*t instead of 3t. Use radians for trigonometric functions. Try smooth functions first. Avoid discontinuities at the limits. Increase Simpson intervals when the function bends sharply.

The theorem is useful in physics, probability, economics, and engineering. Many models describe total change through accumulated rates. A variable boundary may represent time, distance, temperature, or money. This makes the theorem a practical shortcut. It also helps students understand why integration and differentiation are inverse ideas.

Best practice

For best results, compare several nearby x values. Watch how the upper and lower contributions change. The separate terms reveal which moving boundary controls the answer. This can make homework checks clearer. It can also expose sign errors quickly.

If your course uses symbolic notation, treat this answer as a numeric evaluator. It confirms the theorem at one chosen point. It does not replace algebraic simplification. Still, it is helpful for practice. You can test examples, download reports, and compare results before writing final work. It also supports fast review during exam preparation sessions.

FAQs