Question 1017

Definite Integrals: Areas and Volumes

2017 Paper 2 Question 4 Variant

Question

(a)

A flat novelty plate for serving food on is made in the shape of the region enclosed by the curve

{y=x^2 - 6 x + 5}

and the line

{8 y = x - 1.}

Find the area of the plate.

[4]

(b)

A curved container has a flat circular top. The shape of the container is formed by rotating the part of the curve

x=\frac{\sqrt{y}}{a-y^2},

where

{a}

is a constant greater than

{1}

, between the points

{(0,0)}

and

\left( \frac{1}{a-1}, 1 \right)

through

{2\pi}

radians about the

{y\textrm{-axis.}}

(bi)

Find the volume of the container, giving your answer as a single fraction in terms of

{a}

and

{\pi.}

[4]

(bii)

Another curved container with a flat circular top is formed in the same way from the curve

x=\frac{\sqrt{y}}{b-y^2},

and the points

{(0,0)}

and

\left( \frac{1}{b-1}, 1 \right).

It has a volume eight times as great as the container in part (i). Find an expression for

{b}

in terms of

{a.}

[3]