Theory question 111
Integration(a) Evaluate ∫12 x(3x − 2)/(2√x) dx
(b) The equation of a circle is given by 2x2 + 2y2 − 8x + 5y − 10 = 0. Find the: (i) coordinates of the centre; (ii) radius of the circle; (iii) coordinates of P and Q, if the circle cuts the x-axis at the points P and Q.
Show the model answer
Model answer
(a) x(3x − 2)/(2√x) = (3/2)x3/2 − x1/2
∫ = [(3/5)x5/2 − (2/3)x3/2] from 1 to 2
= (12√2/5 − 4√2/3) − (3/5 − 2/3) = 16√2/15 + 1/15 = (16√2 + 1)/15 ≈ 1.575
(b) Divide by 2: x2 + y2 − 4x + (5/2)y − 5 = 0
(i) Centre (2, −5/4)
(ii) r2 = 4 + 25/16 + 5 = 169/16, r = 13/4
(iii) y = 0: x2 − 4x − 5 = 0, (x − 5)(x + 1) = 0; P(−1, 0) and Q(5, 0)
Explanation
Split the fraction into powers of x before integrating; the limits are 1 and 2.