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EN1911 (Mathematics 1A) Numerical Integration Tutorial 1Tutorial - Numerical Integration
Trapeziodal
𝑥𝑛
ℎ
∫ 𝑓(𝑥) 𝑑𝑥 ≈ (𝑦 + 2𝑦1 + 2𝑦2 + ⋯ + 2𝑦𝑛−2 + 2𝑦𝑛−1 + 𝑦𝑛 )
𝑥0 2 0
Simpsons
𝑥𝑛
ℎ 𝑥𝑛 − 𝑥0
∫ 𝑓(𝑥) 𝑑𝑥 ≈ (𝑦 + 4𝑦1 + 2𝑦2 + ⋯ + 2𝑦𝑛−2 + 4𝑦𝑛−1 + 𝑦𝑛 ) where ℎ=
𝑥0 3 0 𝑛
1 𝑑𝑥
1. Approximate ∫0 using the Trapezoidal Rule using 5 data points (step size is 0.25).
1+𝑥 2
2. Repeat Q1 using Simpson’s rule.
1.5
3. Approximate ∫0 𝑥 sin(𝑥) 𝑑𝑥 using the Trapezoidal Rule with step size 0.25.
4. Repeat Q3 using Simpson’s rule.
5. Use Simpson’s rule to approximate the definite integral of the function 𝑓(𝑥) over the
interval 𝑥 = 2 to 𝑥 = 2.6, given the values of the function in the table below.
𝑥 2.0 2.1 2.2 2.3 2.4 2.5 2.6
𝑦 = 𝑓(𝑥) 3.2 2.7 4.1 3.8 3.5 4.6 5.2
6. Use Simpson’s rule with step size ℎ = 0.5 to approximate the integral
3
∫1 √1 + 𝑥 3 𝑑𝑥
1
7. The increase in the voltage on a capacitor is given by 𝑉 = 𝐶 ∫ 𝐼(𝑡) 𝑑𝑡.
For a 10 𝐹 capacitor (recall that F = 10–6F) the following data is recorded for the
current 𝐼(𝑡) against time 𝑡; use this data and Simpson’s rule to estimate the voltage
increase across the capacitor after 2ms.
time (10–3 s) 0.0 0.5 1.0 1.5 2.0
–3
current (10 amp) 2.5 1.75 1.25 0.90 0.70
Answers
1. Sum is 6.2624 Integral is 0.783
2. Sum is 9.4247 Integral is 0.785
3. Sum is 7.1772 Integral is 0.897
4. Sum is 10.6959 Integral is 0.891
5. Sum is 68 Integral is 2.267
6. Sum is 37.3818 Integral is 6.230
7. Sum is 16.3 Integral is 2.7167 (in 𝐶). Divide by 𝐶 = 10 𝐹 to get 𝑉 = 0.272 𝑉
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