5. (a) Find ∫(9x + 6)/x dx, x > 0
(b) Given that y = 8 at x = 1, solve the differential equation
dy/dx = [(9x + 6)1/3]x
giving your answer in the form
y2 = g(x)
6. The area A of a circle is increasing at a constant rate of 1.5 cm2 s–1. Find, to 3 significant figures, the rate at which the radius r of the circle is increasing when the area of the circle is 2 cm2
Figure 2 shows a sketch of the curve C with parametric equations
x = 5t2 - 4, y = t(9 - t2)
The curve C cuts the x-axis at the points A and B.
(a) Find the x-coordinate at the point A and the x-coordinate at the point B.
The region R, as shown shaded in Figure 2, is enclosed by the loop of the curve.
(b) Use integration to find the area of R.
8. (a) Using the substitution x = 2 cos u, or otherwise, find the exact value of
Figure 3 shows a sketch of part of the curve
The shaded region S, shown in Figure 3, is bounded by the curve, the x-axis and the lines with equations x = 1 and x = √2. The shaded region S is rotated through 2π radians about the x-axis to form a solid of revolution.
(b) Using your answer to part (a), find the exact volume of the solid of revolution formed.