Numerical integration Worked Solutions — A-Level Maths

Fully worked, step-by-step solutions to A-Level Numerical integration questions. See exactly how to solve problems on numerical-integration, trapezium-rule, strip-width, ordinates.

numerical-integrationtrapezium-rulestrip-widthordinatesestimateover-under
A-Level70 questionsStep-by-step solutions
Question 1
2 markseasy
The trapezium rule is used to estimate ∫13x dx\int_{1}^{3} \sqrt{x} \, dx with 2 strips. Find the strip width hh.

Worked solution

  1. Identify the limits and number of strips

    a=1, b=3, n=2a=1,\ b=3,\ n=2

    Read these from the integral.

  2. Apply the strip-width formula

    h=b−an=3−12h=\frac{b-a}{n}=\frac{3-1}{2}

    The interval is split into n equal strips.

  3. State the strip width

    h=22=1h=\frac{2}{2}=1

    This is the width of each strip.

Answer
11
Question 2
2 markseasy
The trapezium rule is used to estimate ∫03ex dx\int_{0}^{3} e^{x} \, dx with 3 strips. Find the strip width hh.

Worked solution

  1. Identify the limits and number of strips

    a=0, b=3, n=3a=0,\ b=3,\ n=3

    Read these from the integral.

  2. Apply the strip-width formula

    h=b−an=3−03h=\frac{b-a}{n}=\frac{3-0}{3}

    The interval is split into n equal strips.

  3. State the strip width

    h=33=1h=\frac{3}{3}=1

    This is the width of each strip.

Answer
11
Question 3
2 markseasy
The trapezium rule is used to estimate ∫131x dx\int_{1}^{3} \frac{1}{x} \, dx with 4 strips. Find the strip width hh.

Worked solution

  1. Identify the limits and number of strips

    a=1, b=3, n=4a=1,\ b=3,\ n=4

    Read these from the integral.

  2. Apply the strip-width formula

    h=b−an=3−14h=\frac{b-a}{n}=\frac{3-1}{4}

    The interval is split into n equal strips.

  3. State the strip width

    h=24=0.5h=\frac{2}{4}=0.5

    This is the width of each strip.

Answer
0.50.5
Question 4
2 markseasy
The trapezium rule is used to estimate ∫05x+1 dx\int_{0}^{5} \sqrt{x + 1} \, dx with 5 strips. Find the strip width hh.

Worked solution

  1. Identify the limits and number of strips

    a=0, b=5, n=5a=0,\ b=5,\ n=5

    Read these from the integral.

  2. Apply the strip-width formula

    h=b−an=5−05h=\frac{b-a}{n}=\frac{5-0}{5}

    The interval is split into n equal strips.

  3. State the strip width

    h=55=1h=\frac{5}{5}=1

    This is the width of each strip.

Answer
11
Question 5
2 markseasy
The trapezium rule is used to estimate ∫06x2+1 dx\int_{0}^{6} x^{2} + 1 \, dx with 6 strips. Find the strip width hh.

Worked solution

  1. Identify the limits and number of strips

    a=0, b=6, n=6a=0,\ b=6,\ n=6

    Read these from the integral.

  2. Apply the strip-width formula

    h=b−an=6−06h=\frac{b-a}{n}=\frac{6-0}{6}

    The interval is split into n equal strips.

  3. State the strip width

    h=66=1h=\frac{6}{6}=1

    This is the width of each strip.

Answer
11

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