1.7: Polynomial roots and a volume of revolution

4PM1/2R/November/2020 — Question 10 · 14 marks

1.7 diagram 1
f(x)=32x3−33x+1.f(x)=32x^3-33x+1.
(a) Show that f(1)=0\displaystyle f(1)=0.
(1)
(b) Hence, using an algebraic method, solve f(x)=0\displaystyle f(x)=0.
(4)
The region R\displaystyle R, shown shaded in Figure 4, is bounded by the curve C1\displaystyle C_1 with equation
y=x,y=\sqrt{x},
the curve C2\displaystyle C_2 with equation
y=18x,y=\frac{1}{8x},
and the line with equation x=a\displaystyle x=a.
The curves C1\displaystyle C_1 and C2\displaystyle C_2 intersect at the point B\displaystyle B, with x\displaystyle x coordinate p\displaystyle p, where p<a\displaystyle p<a.
(c) Find the value of p\displaystyle p.
(2)
The region R\displaystyle R is rotated through 360∘\displaystyle 360^\circ about the x\displaystyle x-axis to generate a solid with volume
27π64.\frac{27\pi}{64}.
(d) Use algebraic integration to find the value of a\displaystyle a.
(7)