How Do You Approximate the Area Under a Curve Using the Trapezoidal Rule?

You approximate the area by dividing the interval into equal-width strips, treating each strip as a trapezoid, and adding their areas. This is the trapezoidal rule, covered in IB Mathematics: Applications and Interpretation SL 5.8.

For an interval from $a$ to $b$, divided into $n$ equal strips, the common strip width is

$$h=\frac{b-a}{n}.$$

If the ordinates, or vertical heights, are $y_0,y_1,\ldots,y_n$, the estimated area is

$$A\approx \frac{h}{2}\left[y_0+y_n+2\left(y_1+y_2+\cdots+y_{n-1}\right)\right].$$

The first and last ordinates are used once because they form the two outside edges. Every interior ordinate is shared by two adjacent trapezoids, so each interior value is multiplied by $2$.

Worked Example

Suppose a function has the following values.

There are $3$ strips, not $4$, because four ordinates create three intervals. Their equal width is

$$h=\frac{3-0}{3}=1.$$

Substituting into the complete formula gives

$$A\approx \frac{1}{2}\left[2+8+2(3+5)\right]=13.$$

Therefore, the estimated area under the curve is $13$ square units. Keep the unrounded values in your GDC if the table contains decimals, and round only the final answer as instructed.

The result is an approximation because straight line segments replace parts of the curved graph. A concave-up curve generally produces an overestimate, whereas a concave-down curve generally produces an underestimate. Increasing the number of equal-width strips usually improves the estimate because the trapezoids follow the curve more closely.

A common misconception is that $n$ strips require $n$ ordinates. In fact, $n$ strips require $n+1$ ordinates.

Exam Technique

This is SL content shared by SL and HL and may appear on either GDC-active paper. For the command term estimate, show the calculation of $h$, substitute into the full formula, double every interior ordinate, include squared units, and use $\approx$ rather than $=$. Check that the intervals have equal widths before applying the standard rule.