1.11.2 Integrating standard functions

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83

Question 77

a.

Given that $\tan x$ , use the quotient rule to show that

\frac{dy}{dx} = \sec^2 x

[3]

b.

The cross-section of a high-precision optical lens is modeled by the region bounded by the curve $y = 3\tan^2 x + 2$ , the $x$ -axis, and the vertical boundaries $\frac{\pi}{6}$ and $\frac{\pi}{3}$ .

Show that the area of this cross-section is

3\sqrt{3} - \sqrt{3} - \frac{\pi}{6}

which simplifies to

2\sqrt{3} - \frac{\pi}{6}

Fully justify your answer.

[5]

1.11.2 Integrating standard functions

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83

Question 77

a.

Given that $\tan x$ , use the quotient rule to show that

\frac{dy}{dx} = \sec^2 x

[3]

b.

The cross-section of a high-precision optical lens is modeled by the region bounded by the curve $y = 3\tan^2 x + 2$ , the $x$ -axis, and the vertical boundaries $\frac{\pi}{6}$ and $\frac{\pi}{3}$ .

Show that the area of this cross-section is

3\sqrt{3} - \sqrt{3} - \frac{\pi}{6}

which simplifies to

2\sqrt{3} - \frac{\pi}{6}

Fully justify your answer.

[5]

1.11.2 Integrating standard functions

Question 77

1.11.2 Integrating standard functions Questions

1.11.2 Integrating standard functions

Question 77

1.11.2 Integrating standard functions Questions