Worked example 1
Evaluate a definite integral
Evaluate ∫ from 0 to 2 (3x² + 1) dx
- Integrate to get x³ + x
- Substitute 2: 8 + 2 = 10
- Substitute 0: 0.
- Subtract.
Answer: 10.
Higher Mathematics
Higher Maths Definite integrals and area
Practise definite integrals and area for Scottish Higher Mathematics with worked examples, clear methods and original interactive questions. Evaluate definite integrals and find area between a curve and the x-axis.
Before you start
- Substitute into powers accurately.
- Use brackets with negative limits.
- Know that area is interpreted from accumulated change.
Explanation
A definite integral gives a numerical value between limits. Find an antiderivative F(x), then calculate F(upper) − F(lower).
For area under a curve, check whether the curve is above or below the axis over the interval.
Method and rules
- ∫ from a to b f(x) dx = F(b) − F(a)
- Area under a positive curve equals the definite integral.
- If the curve is below the axis, the integral is negative and area needs its positive value.
Worked examples
Worked example 2
Find area under a curve
Find the area under y = 2x + 1 from x = 1 to x = 3.
- Integrate: x² + x
- Evaluate at 3 and 1.
- Calculate (9 + 3) − (1 + 1)
So: 10 square units.
Worked example 3
Use negative limits
Evaluate ∫ from −1 to 1 x² dx
- Integrate to x³/3
- Evaluate at 1 and −1.
- Use brackets for (-1)³.
Answer:
Watch out
- Subtracting in the wrong order.
- Forgetting brackets around negative limits.
- Calling a negative integral a negative area without interpreting the graph.
Exam reminder
A definite integral is signed. If a question asks for area, check whether any part of the curve lies below the x-axis.
Related Higher Maths topics
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