Theory question 112
Integration(a) Using the trapezium rule with seven ordinates, evaluate ∫03 dx/(x2 + 1), correct to two decimal places.
(b) Using matrix method, solve −2x + y = 3; −x + 4y = 1
Show the model answer
Model answer
(a) h = 0.5; x = 0, 0.5, 1, 1.5, 2, 2.5, 3
y = 1, 0.8, 0.5, 0.3077, 0.2, 0.1379, 0.1
Area ≈ (h/2)[y₀ + y₆ + 2(y₁ + … + y₅)] = 0.25[1.1 + 2(1.9456)] = 0.25 × 4.9912 ≈ 1.25
(b) (−2 1; −1 4)(x; y) = (3; 1); determinant = −8 + 1 = −7
Inverse = −(1/7)(4 −1; 1 −2)
x = −(1/7)(12 − 1) = −11/7, y = −(1/7)(3 − 2) = −1/7
Explanation
Seven ordinates means six strips of width 0.5; for (b) multiply by the inverse matrix.