Evaluate ∫04 √(2x + 1) dx.
- Write √(2x + 1) as (2x + 1)1/2. Its integral is (2x + 1)3/2/(2 × 3/2) = (2x + 1)3/2/3.
- At x = 4: 93/2/3 = 27/3 = 9. At x = 0: 13/2/3 = 1/3.
- 9 − 1/3 = 26/3.
Answer: 26/3 (about 8.67)
Integration
AS Level Mathematics · Pure Mathematics 1. A short explanation of the idea, the rules to remember, the mistake to avoid, a worked example and practice questions with answers.
The idea
Integration is the reverse of differentiation. If you know dy/dx, integrating takes you back to y. Many curves have the same gradient function, so an indefinite integral always ends with a constant, + c. A point on the curve lets you find c.
A definite integral has limits. Put the upper limit and then the lower limit into the integrated expression and subtract. There is no + c.
A definite integral gives the area between a curve and the x-axis. Area below the axis comes out negative, so split the integral where the curve crosses the axis. The same idea gives the area between two graphs and the volume of revolution when a region is turned about an axis.
Rules to remember
- ∫xn dx = xn + 1/(n + 1) + c, for n ≠ −1
- ∫(ax + b)n dx = (ax + b)n + 1/(a(n + 1)) + c, for n ≠ −1
- Definite integral from a to b = (value at x = b) − (value at x = a)
- Area between a curve and the x-axis from x = a to x = b = ∫y dx between a and b
- Area between two graphs = ∫(upper y − lower y) dx between the x-values where they meet
- Volume about the x-axis = π∫y2 dx; volume about the y-axis = π∫x2 dy
Common mistake
When integrating (ax + b)n, students forget to divide by a. ∫(2x + 1)4 dx is (2x + 1)5/10 + c, not (2x + 1)5/5 + c. Differentiate your answer to check it.
Worked example
Practice questions
Try each one, then open the answer.
1. The region bounded by the curve y = x2, the y-axis and the line y = 4 is rotated completely about the y-axis. Find the volume of the solid formed.
- A32π/5
- B16π
- C8π
- D4π
Show answer
Answer: C. Rotating about the y-axis: V = π∫ x2 dy = π∫04 y dy = π[y2/2] from 0 to 4 = 8π. Using the x-axis formula gives 32π/5.
2. Evaluate ∫13 (2x + 1) dx.
- A12
- B10
- C8
- D14
Show answer
Answer: B. ∫(2x + 1) dx = x2 + x. Evaluate: (9 + 3) − (1 + 1) = 12 − 2 = 10.
3. Find ∫(6x2 − 4x + 1) dx.
- A2x3 − 2x2 + x
- B12x − 4
- C2x3 − 2x2 + x + c
- D2x3 − 4x2 + x + c
Show answer
Answer: C. Raise each power by 1 and divide by the new power: 6x3/3 − 4x2/2 + x = 2x3 − 2x2 + x, then add the constant of integration c.
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