TMUA Integration and Area

Rules

$\int x^n\,dx=\dfrac{x^{n+1}}{n+1}+C$ for $n\ne-1$, and you integrate sums term by term. A definite integral $\int_a^b f(x)\,dx=F(b)-F(a)$ gives the signed area.

Area below the axis

Area below the $x$-axis counts as negative in an integral. To find a total area, split at each root and add the positive sizes.

Area between curves

Find where the curves meet, then integrate (upper curve minus lower curve) between those $x$-values.

Checks and shortcuts

Differentiate your answer to check an integral. For areas, sketch first and mark where the curve crosses the axis and which curve is on top.

Over a range symmetric about zero, an odd function such as $x^3$ integrates to zero. The trapezium rule overestimates the area under a curve that bends upwards and underestimates it for a curve that bends downwards.

Rewrite before integrating: $\dfrac{x+1}{\sqrt x}=x^{1/2}+x^{-1/2}$. For $dy/dx=f(x)$ with a given point on the curve, integrate and then use the point to find the constant $C$.

Worked examples

Common mistakes

• Giving the integral when the question asks for area, and so letting positive and negative parts cancel.
• Forgetting to divide by the new power when integrating.
• Subtracting the curves in the wrong order and getting a negative area.
• Differentiating by mistake.

Test yourself on this topic with our free practice questions.