Ask what the graph already tells you
Before integrating, locate the roots and decide where the function is positive or negative. Check for symmetry. An integral can be zero because positive and negative contributions cancel, even when the curve encloses a substantial area.
- Mark roots, signs and the interval of integration on a sketch.
- Decide whether the question asks for an integral or a total geometric area. Split the interval at sign changes when finding area.
- After calculating, check the sign and size against the sketch. Check an antiderivative by differentiating it.
Worked example
Zero integral, positive area
For f(x) = x³ − 4x on [−2, 2], find both the definite integral and the total area between the graph and the horizontal axis.
The roots are −2, 0 and 2. The function is odd: f(−x) = −f(x). Its signed contributions on a symmetric interval cancel, so the integral from −2 to 2 is 0.
On (0, 2), f(x) is negative. The positive area of that half is the integral of 4x − x³ from 0 to 2. An antiderivative is 2x² − x⁴/4, giving 8 − 4 = 4.
The other half has the same area by symmetry. Therefore the total area is 8 square units. The two answers differ because an integral keeps the sign, while geometric area does not.
Your turn
Keep the inner derivative
For g(x) = (2x − 1)⁴, find g′(x) and an antiderivative of g(x).
Show a hint
For the derivative, use the chain rule. For the antiderivative, try a multiple of (2x − 1)⁵ and differentiate to find that multiple.
Show the full solution
The outer derivative contributes 4(2x − 1)³ and the inner derivative contributes 2. Thus g′(x) = 8(2x − 1)³.
Differentiating A(2x − 1)⁵ gives 10A(2x − 1)⁴. Choose A = 1/10. An antiderivative is therefore (2x − 1)⁵/10 + C.
The factor of 2 is multiplied in differentiation and compensated for in integration. Differentiating the final antiderivative checks both the power and the coefficient.
Explain the factor of 2 in each answer. Revisit the worked example and describe the cancellation without using an antiderivative.
What to practise next
Keep differentiation and integration practice separate until the rules are secure. Then practise selecting a method from the graph or algebra.
- Physics & Maths Tutor: integration
Use definite-integral and area questions with the corresponding model answers.
- Physics & Maths Tutor: differentiation
Practise composite functions and chain-rule calculations before mixing them into longer problems.
- Historical MAT practice
Return to a mixed problem and check whether a sketch can save calculation.
The examples and explanations on this page are part of Arij Asad’s teaching guides. Linked papers and worksheets belong to their named creators and publishers. PMT’s topic collections include exam-board questions and other credited materials.