#### To determine

**To:**

Sketch the region and find the enclosed area.

#### Answer

i) The sketch of the region

ii) Area =63

#### Explanation

**1) Concept:**

Formula:

The area A of the region bounded by the curves y=f(x), y=g(x) and the lines x=a and x=b is

A= ∫abfx-gxdx

fx-gx=fx-gx when fx≥g(x)gx-fx when gx≥f(x)

**2) Given:**

y=sec2x and

y=8cosx, -π3≤x≤π3

**3) Calculation:**

Here,8cosx≥sec2x when -π3≤x≤π3, therefore let’s assume that

fx=8cosx

gx=sec2x

Therefore, the required area is

A=∫-π3π38cosx-sec2xdx

A=∫-π3π38cosx-sec2x dx

Using thesum and the difference rule and the constant multiple rule of integral it becomes

A=8∫-π3π3cosxdx-∫-π3π3sec2xdx

Compute the integral using the standard integration rule.

A=8sinx-π3π3-tanx-π3π3

Evaluate the integral by plugging the upper and the lower limits of integration.

A=8sinπ3-sin-π3-tanπ3-tan-π3

As sinx and tanx are the odd functions, f-x=-fx

A=8sinπ3+sinπ3-tanπ3+tanπ3

A=832+32-3+3

A=8·3-23

A=63

**Conclusion:**

i) The sketch of the region

ii) Area =63