(a) Evaluate : (int_{1} ^{4} frac{x(3x – 2)}{2sqrt{x}} mathrm {d} x)
(b) The equation of a circle is given by (2x^{2} + 2y^{2} – 8x + 5y – 10 = 0). Find the :
(i) coordinates of the centre ; (ii) radius of the circle .
Explanation
(a) (int_{1} ^{4} frac{x(3x -2)}{2sqrt{x}} mathrm {d} x)
= (int_{1} ^{4} frac{1}{2} x(x^{-frac{1}{2}})(3x – 2) mathrm {d} x)
= (frac{1}{2} int_{1} ^{4} (3x^{frac{3}{2}} – 2x^{frac{1}{2}}) mathrm {d} x)
= (frac{1}{2} [frac{3x^{frac{5}{2}}}{frac{5}{2}} – frac{2x^{frac{3}{2}}}{frac{3}{2}}]_{1} ^{4})
= (frac{1}{2} [frac{6x^{frac{5}{2}}}{5} – frac{4x^{frac{3}{2}}}{3}]_{1} ^{4})
= (frac{1}{2} [frac{6(4^frac{5}{2})}{5} – frac{4(4^frac{3}{2})}{3}] – frac{1}{2} [frac{6(1^frac{5}{2})}{5} – frac{4(1^frac{3}{2})}{3}])
= (frac{1}{2} [frac{6 times 32}{5} – frac{4 times 8}{3}] – frac{1}{2} [frac{6}{5} – frac{4}{3}])
= (frac{1}{2}[38.4 – 10.67] – frac{1}{2} [1.2 – 1.33])
= (frac{1}{2} [27.73 + 0.13])
= (13.93)
(b) (2x^{2} + 2y^{2} – 8x + 5y – 10 = 0)
(x^{2} + y^{2} – 4x + frac{5}{2}y = 5)
(x^{2} – 4x + [frac{1}{2} (-4)]^{2} + y^{2} + frac{5}{2}y + [frac{1}{2} (frac{5}{2})]^{2} = 5 + 4 + frac{25}{16} = frac{169}{16})
((x – 2)^{2} + (y + frac{5}{2})^{2} = frac{169}{16})
(i) coordinates of the centre = ((2, -frac{5}{2}))
(ii) Radius = (sqrt{frac{169}{16}} = frac{13}{4})