TMUA: Definite integrals and area
Definite integrals and areas.
- Section 1 Part 1 · MM7 Integration · spec MM7.1-MM7.2
- Papers 1 and 2
- 7 practice questions
- No calculator
What the specification covers
- Integral vs area when the curve crosses the axis
- Simplify before integrating
Key ideas
- \(\displaystyle\int_a^bf(x)\,dx=F(b)-F(a)\).
- Regions below the axis give negative integrals: the area is the sum of the absolute values of the pieces.
- Find where the curve crosses the axis inside the interval and split the integral there.
Common mistakes
- 'The integral from 0 to 6' and 'the area between 0 and 6' can be different numbers.
- Signs: \(F(0)-F(4)\) versus \(F(4)-F(0)\).
Exam tip
Sketch the curve first to see which parts are below the axis.
Worked example
Worked example
What is the total area enclosed between the curve \(y=x^2-4x\), the \(x\)-axis and the lines \(x=0\) and \(x=6\)?
- \(\frac{128}{3}\)
- \(0\)
- \(\frac{64}{3}\)
- \(\frac{32}{3}\)
- \(16\)
Answer: C: \(\frac{64}{3}\)
The curve is below the axis on \(0
Practice questions
Try each question before opening the solution. No calculator: the TMUA does not allow one. Every answer here was re-checked independently by computer before it was published.
Question 1
What is the total area enclosed between the curve \(y=x^2-3x\), the \(x\)-axis and the lines \(x=0\) and \(x=4\)?
- \(\frac{8}{3}\)
- \(\frac{9}{2}\)
- \(- \frac{8}{3}\)
- \(\frac{19}{3}\)
- \(\frac{11}{6}\)
Show the answer and solution
Answer: D: \(\frac{19}{3}\)
The curve is below the axis for \(0
Question 2
What is the total area enclosed between the curve \(y=x^2-6x\), the \(x\)-axis and the lines \(x=0\) and \(x=7\)?
- \(\frac{118}{3}\)
- \(36\)
- \(\frac{10}{3}\)
- \(- \frac{98}{3}\)
- \(\frac{98}{3}\)
Show the answer and solution
Answer: A: \(\frac{118}{3}\)
The curve is below the axis for \(0
Question 3
What is the total area enclosed between the curve \(y=x^2-5x\), the \(x\)-axis and the lines \(x=0\) and \(x=7\)?
- \(\frac{67}{2}\)
- \(\frac{125}{6}\)
- \(- \frac{49}{6}\)
- \(\frac{38}{3}\)
- \(\frac{49}{6}\)
Show the answer and solution
Answer: A: \(\frac{67}{2}\)
The curve is below the axis for \(0
Question 4
What is the total area enclosed between the curve \(y=x^2-5x\), the \(x\)-axis and the lines \(x=0\) and \(x=6\)?
- \(-18\)
- \(\frac{125}{6}\)
- \(\frac{71}{3}\)
- \(18\)
- \(\frac{17}{6}\)
Show the answer and solution
Answer: C: \(\frac{71}{3}\)
The curve is below the axis for \(0
Question 5
What is the total area enclosed between the curve \(y=x^2-6x\), the \(x\)-axis and the lines \(x=0\) and \(x=8\)?
- \(- \frac{64}{3}\)
- \(\frac{64}{3}\)
- \(\frac{152}{3}\)
- \(\frac{44}{3}\)
- \(36\)
Show the answer and solution
Answer: C: \(\frac{152}{3}\)
The curve is below the axis for \(0
Question 6
What is the total area enclosed between the curve \(y=x^2-4x\), the \(x\)-axis and the lines \(x=0\) and \(x=5\)?
- \(\frac{7}{3}\)
- \(\frac{25}{3}\)
- \(13\)
- \(- \frac{25}{3}\)
- \(\frac{32}{3}\)
Show the answer and solution
Answer: C: \(13\)
The curve is below the axis for \(0
Keep going
- Previous topic: Tangents, normals and stationary points
- Next topic: Integration: the Fundamental Theorem and the trapezium rule
- All TMUA topics · Timed TMUA paper simulator · Official TMUA past papers and specimen papers
TMUA is run by UAT-UK. A Level Math Revision is independent: it is not affiliated with or endorsed by UAT-UK, Pearson VUE or any university. Every question here is our own.