How to Find a Definite Integral on a TI-84 with fnInt(

A definite integral gives the signed area trapped between a curve and the x-axis over a fixed interval, and the TI-84 Plus and CE compute it numerically with MATH ▸ 9:fnInt(. You type the function, tell the calculator which variable to use, and give the two limits; it returns a single number.

Because the tool is numeric, it answers “how much area” rather than “what is the antiderivative”. This page sets out the four arguments, works through an example you can check by hand, explains why a negative answer is meaningful, and describes the accuracy you can expect and where it breaks down.

What fnInt( works out

fnInt( chops the interval between the two limits into many thin strips, works out the area of each, and adds them up. This is numerical quadrature, and for a smooth curve it gets very close to the exact value. It does not produce an antiderivative, so it cannot help with an indefinite integral — only with the area over a stated interval.

If you want the graphical version instead, the same idea is available from the graph screen. With the curve in Y1, press 2nd TRACE for the CALC menu and choose 7:∫f(x)dx, then set the lower and upper limits with the cursor. The calculator shades the region as you move.

The four arguments

Press MATH, scroll to 9:fnInt(, and press ENTER to paste it. The command takes four parts:

fnInt(expression, X, lower, upper)

1. the expression to integrate,
2. the variable X, typed with the X,T,θ,n key,
3. the lower limit,
4. the upper limit.

The limits run lower first, upper second. Swapping them flips the sign of the result, which is mathematically correct but rarely what you want. A stray comma in the wrong place raises ERR: SYNTAX; the SYNTAX error page shows how to find the offending character.

A worked example

Find the area under y = x² between x = 1 and x = 4. Enter:

fnInt(X²,X,1,4)

and press ENTER.

By hand, the antiderivative of x² is x³ ÷ 3. At the upper limit that is 64 ÷ 3, and at the lower limit it is 1 ÷ 3, so the difference is 63 ÷ 3 = 21. The calculator returns 21 exactly, which is a good sign that the syntax was right and the function was entered as intended.

The result is a signed area

The number fnInt gives is not a plain amount of space under the curve. It is a signed area: any part of the curve below the x-axis contributes a negative amount, and any part above contributes a positive amount. The final figure is the balance of the two.

For instance, over the interval from 0 to 1, the line y = x − 1 spends the whole time below the axis. Enter:

fnInt(X-1,X,0,1)

The answer is −0.5. A negative result always means the region below the axis is larger than the region above it — it does not mean you made a mistake.

The accuracy note

fnInt is an approximation, not an exact answer. For smooth functions the default setting is accurate to several decimal places, and for a polynomial like x² the result is exact because the method handles it cleanly. Accuracy can drop when the curve has a sharp corner, a steep spike, or a genuine discontinuity inside the interval.

There is an optional tolerance argument at the end of the command, fnInt(expression, X, lower, upper, tolerance), which lets you ask for a finer calculation. Making it smaller improves the answer for a well-behaved function, but pushing it too far is pointless: the calculator’s internal precision eventually limits the gain, and a very small tolerance only costs more time. Leave it out unless a question specifically needs extra decimal places.

Functions that are undefined inside the interval are the real trap. Integrating 1 ÷ x across x = 0, for example, samples values on both sides of a break that should not be crossed; the calculator will return a number, but it is not a true area, and you may instead meet ERR: DOMAIN at the troublesome point, as described on the DOMAIN error page.

Where the calculator stops short

The method confirms a value; it does not earn method marks. When a question says “evaluate” or “show that”, the expected route is the antiderivative written out, with fnInt used afterwards as a check. Reach for the algebraic route whenever the question asks for an indefinite integral, a substitution, or a proof — none of those are numerical problems.

The companion derivative tool is covered on our derivatives page, and the question of what a function tends towards at a point, rather than what it reaches, is the subject of limits.

On the AP Calculus exams a graphing calculator such as the TI-84 Plus or Plus CE is allowed on the sections that permit one, though a calculator with a computer algebra system is prohibited. The rules are revised from time to time, so confirm the current wording with the AP calculator policy and your teacher.