Integration and applications of integration — Question 17

QCAA 2021, Paper 1 · 7 marks

Q17 · 2021 · Technology-freeComplex familiar7 marks

QUESTION 17 (7 marks)

The area between the graphs of the functions

y=4x

and

y=2x^2

is rotated about the

y

-axis to form a solid of revolution with a volume of

V

units

^3

. Determine the exact value of

V

WORKED SOLUTION

7 marks

ANSWER

V=\dfrac{16\pi}{3}

units

^3

Worked solution

The curves intersect at

(0,0)

and

(2,8)

. In terms of

y

x=y/4

for the line and

x=\sqrt{y/2}

for the parabola. Using washers about the

y

-axis,

V=\pi\int_0^8\left(\frac y2-\frac{y^2}{16}\right)dy=\frac{16\pi}{3}.

correctly uses simultaneous equations to establish an equation in one unknown

[1 mark]

correctly determines

y

-coordinates of the points of intersection

[1 mark]

correctly determines functions in the form

x=f(y)

[1 mark]

determines expression to represent the volume between the two curves

[1 mark]

integrates expression

[1 mark]

determines positive value of

V

in terms of

\pi

[1 mark]

shows logical organisation, communicating key steps to at least the start of finding the volume of revolution

[1 mark]

One QCAA sample method typeset for web; criterion wording is adapted from the official marking guide.

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