Calculate the area of the shape bounded by lines y = x ^ 2-4x + 5 y = x + 1

We calculate the area of the figure bounded by the lines y = x ^ 2 – 4 * x + 5 and y = x + 1.

The graph of the first function is a parabola, the branches of which are directed upwards. A straight line graph is a straight line.

x + 1 – (x ^ 2 – 4 * x + 5) = x + 1 – x ^ 2 + 4 * x – 5 = -4 + 5 * x – x ^ 2 = -x ^ 2 + 5 * x – 4;

S = (1 to 4) ∫ (-x ^ 2 + 5 * x – 4) dx = (1 to 4) (-x ^ 3/3 + 5 * x ^ 2/2 – 4 * x) = (1 to 4) (-1/3 * x ^ 3 + 5/2 * x ^ 2 – 4 * x) = (-1/3 * 4 ^ 3 + 5/2 * 4 ^ 2 – 4 * 4 ) – (-1/3 * 1 ^ 3 + 5/2 * 1 ^ 2 – 4 * 1) = (-64/3 + 40 – 16) – (-1/3 + 5/2 – 4) = – 64/3 + 24 + 1/3 – 5/2 + 4 = -63/3 + 28 – 5/2 = -21 + 28 – 2.5 = 7 – 2.5 = 4.5.