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.
Back to Higher Mathematics
Higher Mathematics
Definite integrals and area
Evaluate integrals between limits and interpret positive area under a curve.
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: 23
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.