Mathematical Methods — Question 11

QCAA 2021, Paper 2 · 5 marks

Q11 · 2021 · Technology-activeSimple familiar5 marks

QUESTION 11 (5 marks)

Consider the function

f(x)=e^x\sin(x)

0\le x\le2\pi

State the exact values of the

x

-intercepts of the graph of

f(x)

[2 marks]

Write an expression for the area enclosed between the graph of

f(x)

and the

x

-axis.

[2 marks]

Determine the area enclosed between the graph of

f(x)

and the

x

-axis to the nearest square unit.

[1 mark]

WORKED SOLUTION

5 marks

ANSWER

x=0,\pi,2\pi

; b)

\int_0^\pi e^x\sin x\,dx+\left|\int_\pi^{2\pi}e^x\sin x\,dx\right|

; c) 291 square units

Worked solution

a) At an

x

-intercept,

0=e^x\sin x

. Since

e^x\ne0

\sin x=0

, so

x=0,\pi,2\pi

\int_0^\pi e^x\sin x\,dx+\left|\int_\pi^{2\pi}e^x\sin x\,dx\right|.

c) Using a GDC, the enclosed area is

291

square units to the nearest unit.

correctly generates the equation required

[1 mark]

correctly determines the three x-intercepts

[1 mark]

determines expression for integral above the x-axis

[1 mark]

determines expression for integral below the x-axis (including absolute value brackets)

[1 mark]

determines area to the nearest unit

[1 mark]

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

How many marks did you earn?

Compare your working with the guide above.