How Do You Calculate a Definite Integral Using Technology?

Use your graphing display calculator (GDC) to enter the integrand, lower bound, and upper bound into its numerical integration function. In an IB Mathematics: Analysis and Approaches exam, write the definite integral first, then report the calculator result, normally to three significant figures.

The Method

A definite integral has the form

$$\int_{a}^{b} f(x)\,dx,$$

where $f(x)$ is the integrand, $a$ is the lower bound, and $b$ is the upper bound. On most GDCs, select the numerical integration command and enter these four items:

1. the function $f(x)$;
2. the variable $x$;
3. the lower bound $a$;
4. the upper bound $b$.

For example, calculate

$$\int_{0}^{1} e^{-x^2}\,dx.$$

This integral does not have an elementary antiderivative, so numerical technology is appropriate. Enter $e^{-x^2}$ with lower bound $0$ and upper bound $1$. The GDC gives

$$\int_{0}^{1} e^{-x^2}\,dx \approx 0.746824\ldots,$$

so the exam answer is

$$\boxed{0.747}$$

to three significant figures.

Before accepting the result, graph the integrand over the stated interval. This helps detect entry errors and shows whether cancellation occurs. If the limits are reversed, use $\int_{b}^{a} f(x)\,dx=-\int_{a}^{b} f(x)\,dx$, so the sign changes while its magnitude remains the same for the same integrand and bounds.

The integral represents signed area. Regions above the $x$-axis contribute positively, while regions below it contribute negatively. If a question asks for total area and the graph crosses the axis, find the zeros and split the interval, integrating each region with a positive contribution.

A common misconception is that a calculator result alone earns full marks. It may not: IB expects the correct integral expression before its numerical evaluation. Also check that the bounds are entered in the correct order and that the calculator has not confused $x$ with another variable.

Exam Technique

This is SL 5.5: Introduction to integration and is normally assessed using technology on Paper 2. Write the setup, state that technology was used if helpful, and give an exact answer or a numerical answer to three significant figures unless the question specifies otherwise.