Integration — common mistakes

If a question asks for the 'total area' of a region that is partly above and partly below the x-axis, you must calculate two separate definite integrals. Find the x-intercept where the curve crosses the axis, split the integral at that point, and add the absolute values of the two results. A single integral across the whole region will give the 'net area', which might be incorrect.

An indefinite integral, $\int f(x) \, dx$, gives a function, $F(x) + C$, representing a family of curves. A definite integral, $\int_a^b f(x) \, dx$, gives a single numerical value, representing the net area under the curve between $x=a$ and $x=b$.

If the curve $y=f(x)$ is below the x-axis between $x=a$ and $x=b$, the definite integral $\int_a^b f(x) \, dx$ will be negative. The geometric area is the absolute value (the positive magnitude) of this result.

The power rule for $x^n$ is $\frac{x^{n+1}}{n+1} + C$. For $1/x = x^{-1}$, we have $n=-1$. Applying the rule would lead to division by $n+1 = -1+1 = 0$, which is undefined. The integral of $1/x$ is a different function, $\ln|x|$, which is covered in the A2 syllabus (Paper 3).

To find the area between two curves, $y=f(x)$ and $y=g(x)$, from $x=a$ to $x=b$, where $f(x)$ is the upper curve, you calculate the integral of the difference: Area $= \int_a^b (f(x) - g(x)) \, dx$. The limits $a$ and $b$ are often the x-coordinates of the points where the curves intersect.