Fundamental Theorem of Calculus Calculator

Calculator Inputs

Function f(x) Use x, pi, e, +, -, *, /, ^, and functions like sin(x).

Lower limit a

Upper limit b

Point x0 for FTC check

Numerical intervals

Decimal places

Integration method

Primary output

Formula Used

Part one of the theorem uses this relation:

∫ from a to b of f(x) dx = F(b) - F(a), where F'(x) = f(x).

Part two uses the accumulation rule:

If A(x) = ∫ from a to x of f(t) dt, then A'(x) = f(x).

The average value is found with:

Average value = (1 / (b - a)) × ∫ from a to b of f(x) dx.

How To Use This Calculator

Enter a function using x as the variable.
Add the lower and upper limits for the integral.
Enter x0 to test the accumulation derivative rule.
Select a numerical method and interval count.
Choose the main result type, then press Calculate.
Download the CSV or PDF report after reviewing results.

Example Data Table

Function	a	b	x0	Expected Idea
x^2	0	3	2	Integral is 9, and A'(2) is 4.
sin(x)	0	pi	1	Integral is near 2.
3*x^2 + 1	1	4	2	Net change equals the integral of the rate.
exp(x)	0	1	0.5	Average value uses the integral over the interval length.

Fundamental Theorem of Calculus Calculator Guide

The fundamental theorem links area and change. It says that integration can reverse differentiation. It also says an accumulation function has an instant rate. This calculator uses those two ideas in a practical way. You enter a function, a lower limit, an upper limit, and a checking point. The tool then estimates the definite integral. It also shows net change, average value, and the derivative of an accumulation curve.

Why This Calculator Helps

Manual integration can be slow. Some functions are hard to integrate by hand. Numerical methods give a careful estimate when a symbolic antiderivative is unknown. This page supports common functions, powers, constants, and several integration methods. It is useful for homework checking, class examples, and applied modeling. You can compare Simpson, trapezoid, and midpoint estimates. More intervals usually improve the result.

Understanding The Two Parts

Part one connects a definite integral to an antiderivative. If F is an antiderivative of f, then the signed area from a to b equals F(b) minus F(a). Part two describes accumulation. If A(x) equals the integral from a to x of f(t), then A prime at x equals f(x). This means the rate of accumulated area is the original function value.

Best Use Cases

Use the calculator for distance from velocity, growth from a rate, charge from current, or total cost from marginal cost. It also helps when you need average value over an interval. The sign of the answer matters. A negative integral means the curve has more signed area below the axis. A positive result means more signed area above the axis.

Accuracy Tips

Use radians for trigonometric inputs. Place multiplication signs between factors. Write 2*x instead of 2x for the clearest parsing. Avoid discontinuities inside the interval. If a function explodes near a limit, a numeric answer may be unstable. Increase intervals and compare methods. Agreement across methods gives better confidence. Always review the expression, bounds, and chosen method before exporting reports.

Export And Review

The export buttons save the current calculation. Use CSV for spreadsheets. Use PDF for simple sharing. Keep one record for each method tested. Accuracy notes help later review and audit checks.

FAQs

What does this calculator find?

It estimates definite integrals, net change, average value, and accumulation derivative checks using the fundamental theorem of calculus.

Does it find symbolic antiderivatives?

No. It uses numerical integration. This makes it useful for many functions, even when a simple antiderivative is not available.

Which method should I choose?

Simpson rule is a strong default for smooth functions. Trapezoid and midpoint are useful for comparison and checking stability.

Why must Simpson intervals be even?

Simpson rule works on pairs of subintervals. If you enter an odd count, the calculator raises it by one automatically.

Can I use trigonometric functions?

Yes. Use sin(x), cos(x), tan(x), and inverse functions. Trigonometric inputs are handled in radians.

What does net change mean?

Net change is the accumulated result of a rate function over an interval. It equals the definite integral of that rate.

What is the accumulation derivative check?

It compares A'(x0) for A(x)=∫a to x f(t)dt with f(x0). The values should be very close for smooth functions.

Can I export my result?

Yes. After calculation, use the CSV button for spreadsheet data or the PDF button for a simple report.