Theory question 112
Integration(a) Use the trapezium rule with five ordinates to evaluate ∫01 3dx/(1 + x2) correct to four significant figures.
(b) If A = (2 1; 3 2) find the image of the point (1, 2) under the linear transformation A2 + A + 2I, where I is the 2 x 2 unit matrix.
Show the model answer
Model answer
(a) h = 0.25; ordinates at x = 0, 0.25, 0.5, 0.75, 1: 3, 2.8235, 2.4, 1.92, 1.5
Area ≈ ½(0.25)[3 + 1.5 + 2(2.8235 + 2.4 + 1.92)] = 2.348
(b) A2 = (7 4; 12 7)
A2 + A + 2I = (11 5; 15 11)
Image of (1, 2) = (11 + 10, 15 + 22) = (21, 37)
Explanation
Five ordinates means four strips of width 0.25. Matrices are written (row 1; row 2).