Integration — common mistakes

In trapezium rule questions, be very careful with the formula. A common mistake is to forget to multiply the sum of the 'middle' y-values by 2. Also, ensure your calculator is in radians mode if the function involves trigonometric terms with angles in radians. State the number of decimal places or significant figures required by the question.

When finding the area between two curves, it's essential to correctly identify which function is 'upper' and which is 'lower' within the integration interval. If you get them the wrong way around, your answer will be negative. You can simply take the absolute value, but it's better to be sure from the start by sketching the graphs or testing a point.

An indefinite integral, $\int f(x) \, dx$, gives a general function (a family of curves) and always includes a constant of integration, $+c$. A definite integral, $\int_a^b f(x) \, dx$, has limits and evaluates to a specific number, representing the signed area under the curve between those limits.

In questions asking for a general solution or an indefinite integral, forgetting '+c' will typically lose you one accuracy mark. For definite integrals, the '+c' is not needed as it cancels out, so you won't lose marks there.

A negative result means that there is more area below the x-axis than above it in the given interval. If the question asks for the 'area', which is a geometric quantity and must be positive, you need to re-evaluate. This usually involves splitting the integral at the roots and taking the absolute value of the parts that are below the axis.

The question will explicitly tell you to use the trapezium rule. It's used for functions that are difficult or impossible to integrate analytically, or when you are given a table of values instead of a function.