How To Do Integrals On Ti-84

How to Do Integrals on a TI‑84 Calculator

Learning how to perform integrals on a TI‑84 Plus (or TI‑84 Plus CE) is a practical skill for anyone studying calculus, physics, or engineering. The calculator can evaluate both definite and indefinite integrals numerically, and it also lets you visualize the area under a curve. Below you’ll find a step‑by‑step guide, the underlying mathematical reasoning, common pitfalls, and a FAQ section to help you master integration on the TI‑84.

Why Use the TI‑84 for Integration? * Speed – Numerical integration is instantaneous for most functions.

• Verification – You can check hand‑computed results quickly. * Visualization – The graphing feature lets you see the region whose area you’re calculating.
• Accessibility – No need for software installation; the calculator is always at hand.

The TI‑84 does not produce symbolic antiderivatives (like ∫ x² dx = x³/3 + C), but its fnInt function computes definite integrals to a high degree of accuracy using adaptive Simpson’s rule.

Getting Ready: Mode and Settings

Before you start, ensure the calculator is in the correct mode:

1. Press [MODE].
2. Highlight Float (or a specific number of decimal places if you prefer).
3. Make sure Radian is selected if you’re working with trigonometric functions; otherwise choose Degree for degree‑based problems.
4. Press [ENTER] to confirm each setting, then [2ND] → [MODE] (QUIT) to return to the home screen.

Step‑by‑Step: Computing a Definite Integral

The primary tool for integration on the TI‑84 is the fnInt function, located under the MATH menu.

1. Access fnInt

• Press [MATH].
• Scroll down to 9:fnInt( (or press the 9 key directly).
• The screen will show fnInt(.

2. Enter the Function

• Type the integrand exactly as you would in the Y= editor.
  • Example: to integrate x^2, type X^2.
  • Use the [X,T,θ,n] button for the variable X.
• After the function, press [,] (comma).

3. Specify the Variable of Integration

• Press [X,T,θ,n] again to indicate the variable (usually X).
• Press [,].

4. Input the Lower Limit

• Type the lower bound (a number or expression).
• Press [,].

5. Input the Upper Limit * Type the upper bound.

• Close the parenthesis with [)].

6. Execute

• Press [ENTER].
• The calculator returns a numerical approximation of the definite integral.

Example: Compute ∫₀² (3x + 1) dx

MATH → 9:fnInt( → 3X+1 → , → X → , → 0 → , → 2 → ) → ENTER

Result: 8.0000 (the exact value is 8).

Visualizing the Integral

Seeing the area under the curve reinforces the concept.

1. Press [Y=] and enter the function in Y1 (e.g., 3X+1).
2. Press [WINDOW] and set appropriate Xmin, Xmax, Ymin, Ymax to capture the region.
3. Press [2ND] → [TRACE] (CALC).
4. Choose 7:∫f(x)dx.
5. Enter the lower limit, press [ENTER], then the upper limit, press [ENTER].
6. The calculator shades the area and displays the numerical value at the bottom of the screen.

Handling Improper Integrals

The TI‑84 cannot directly evaluate integrals with infinite limits, but you can approximate them by using a large finite bound.

• Example: Approximate ∫₀^∞ e^(–x) dx.
• Choose a large upper bound, say 10: fnInt(e^(-X),X,0,10).
• Result ≈ 0.9999546, which is close to the exact value 1. Increase the bound for better accuracy.

Dealing with Piecewise Functions If your integrand changes definition over the interval, split the integral at the breakpoints.

• Suppose f(x) = { x² for x < 1, 2x for x ≥ 1 } and you need ∫₀² f(x) dx.
• Compute two separate integrals:
  • fnInt(X^2,X,0,1)
  • fnInt(2*X,X,1,2)
• Add the results together.

Common Mistakes and How to Avoid Them

Scientific Explanation: How fnInt Works

The fnInt command implements an adaptive Simpson’s rule, a numerical integration technique that:

1. Divides the interval [a, b] into subintervals.
2. Applies Simpson’s quadratic approximation on each subinterval.
3. Estimates the error by comparing results with coarser and finer partitions.
4. Refines subintervals where the error estimate exceeds a tolerance (default ≈ 1 × 10⁻⁵).
5. Sums the contributions to produce the

the integral approximation by adding the weighted area estimates from all subintervals. Because Simpson’s rule uses parabolic segments, it achieves high accuracy for smooth functions while requiring relatively few function evaluations. The adaptive nature of the algorithm concentrates more points where the function changes rapidly (e.g., near peaks or discontinuities) and uses fewer points where the function is nearly linear, thereby balancing speed and precision.

Practical Tips for Getting the Most Out of fnInt

1. Increase Precision When Needed – Press [MODE], scroll to Float, and select a higher number of decimal places (e.g., 6 or 8) before running fnInt. The calculator will display more digits of the result.
2. Check Convergence – For improper integrals, compute the value with two different large bounds (say B and 2B). If the results differ by less than your desired tolerance, the approximation is likely adequate.
3. Use the Graph to Verify – After shading the area with the CALC menu, visually inspect whether the shaded region matches the expected shape. If the shading looks truncated, adjust the window or split the interval.
4. Avoid Over‑loading Memory – Very tight tolerances (e.g., setting fnInt via the ∫f(x)dx option with an extremely small error tolerance) can cause the calculator to spend excessive time refining subintervals. Stick to the default tolerance unless you have a specific reason to change it.
5. Combine with Symbolic Checks – When possible, compare the numerical result with an antiderivative evaluated at the limits (using the calculator’s nDeriv or manual algebra) to catch gross errors.

Example: Numerical Approximation of a Challenging Integral

Compute ∫₀^{π/2} sin(x²) dx (a Fresnel‑type integral).

1. Enter fnInt(sin(X^2),X,0,π/2).
2. The TI‑84 returns ≈ 0.310268.
3. Increasing the upper bound to 2 (i.e., fnInt(sin(X^2),X,0,2)) changes the result only in the sixth decimal place, confirming stability.

Conclusion

The TI‑84 Plus’s fnInt function provides a reliable, user‑friendly way to obtain numerical definite integrals, handling proper integrals directly and allowing approximations for improper or piecewise cases through simple bounds or interval splitting. By understanding its adaptive Simpson’s‑rule basis, watching for common pitfalls, and applying the tips above, you can confidently use the calculator for both routine coursework and more advanced applications where an exact antiderivative is unavailable or cumbersome to derive. Always remember that the output is a decimal approximation; interpret it within the context of the problem’s required precision, and verify results with graphical or alternative numerical checks when accuracy is critical.