Definite Integrals on a Casio fx-9750GII

Confirm that the task is a numerical definite integral

A numerical definite integral has both limits of integration and produces a value for that interval. For example, the integral of x² from 0 to 2 is 8/3. An indefinite integral asks for a family of antiderivatives, x³/3+C, which is a different output. Do not expect a numerical integration command to return the symbolic formula with C. Confirm the exact model printed on the calculator before following menu labels, because another Casio family can use a template or different key sequence.

Open the documented integration command

In the fx-9750GII calculation mode, open OPTN, select CALC, then the integration item marked ∫dx. The manual's linear syntax takes the function in X, lower bound a, upper bound b and optional tolerance: ∫(f(X),a,b,tol). Use the calculator's X variable key and the displayed separators. The optional tolerance defaults to 10^−5 when omitted. These are fx-9750GII software-manual instructions; a different model's natural-display template should be checked in its own manual.

Work a polynomial example before a complex expression

Enter the function X² with lower bound 0 and upper bound 2, then execute the calculation. The expected result is approximately 2.6666667 because the antiderivative X³/3 gives 8/3−0. This example has simple syntax and a known exact answer, so it checks the meaning of the bounds without relying on a difficult integrand. In the linked browser form, use x^2 with bounds 0 and 2 to verify the same mathematical value through its separate numerical method.

Use radians for trigonometric integration

The Casio manual specifies Rad mode for trigonometric integration. For the integral of sin(X) from 0 toπ, use a radian setting and the calculator's πconstant; the exact answer is 2 because−cos(π)+cos(0)=2. The browser check also uses radian trigonometric functions; enter sin(x), lower bound 0 and upper bound 3.141592653589793. Changing the angle convention changes the function being integrated, so selecting radians is part of defining the expression rather than merely choosing a display style.

Track signs when a curve crosses the axis

For the integral of X from−2 to 2, the signed result is zero. The triangle below the axis contributes −2 and the triangle above contributes 2. The total geometric area is 4, obtained by adding magnitudes or integrating |X|. A zero result is therefore reasonable for signed accumulation. If a problem asks for distance or total area, identify sign changes and formulate the appropriate positive contributions rather than assuming the calculator has selected that interpretation automatically.

Split intervals for a clearer numerical check

For the integral of X² from 0 to 2, splitting at 1 gives the sum of 1/3 and 7/3, again 8/3. This verifies the interval-addition property and can help isolate an error in a more complicated input. Splitting at known changes in behaviour can also help numerical work, although agreement alone is not a rigorous error bound. Use exact split points from the problem where possible, and retain enough intermediate digits so that rounding each partial result does not dominate the final sum.

Do not interpret tolerance as a universal accuracy promise

A numerical integrator can struggle with sharp peaks, discontinuities, rapid oscillation or singular endpoints. The Casio guide discusses timeout and interval-splitting considerations; requesting a smaller tolerance does not change an invalid mathematical problem into a valid one. For 1/X across zero, an ordinary integral is not defined, and a symmetric cancellation is not proof of convergence. For 1/sqrt(X) starting at zero, an improper-integral analysis is needed even though the integral converges mathematically.

Compare the mathematical value rather than identical screens

The handheld and browser use their own numerical procedures and formatting, so final displayed digits can differ. For the two checks here, compare against 8/3 and 2, which come from antiderivatives. The local form uses Simpson’s rule with 1000 subintervals and does not accept a user tolerance or output a symbolic primitive. Record the function, bounds, angle convention and model when documenting a Casio calculation so that a later reader can reproduce the intended problem without guessing hidden settings.

Related guides

Calculate a Definite Integral Online and Check Its Meaning

Enter a function and finite bounds, verify a numerical integral with antiderivatives, distinguish signed accumulation from area, and recognize method limits.

Derivative Formulas with Worked Numerical Examples

Apply power, product, quotient and chain rules, distinguish a derivative formula from a slope at a point, and check two examples numerically.

Scientific Notation on the Casio fx-9750GII

Use the fx-9750GII Sci and Norm display settings, distinguish EXP from a power key, and check large and small numbers with worked conversions.