#### To determine

**To sketch:**

The region bounded by the given and find the area of the region

#### Answer

The sketch of the region

Area=2-2ln(2)

#### Explanation

**1) Concept:**

Area is the integral of the difference of two functions.

2) **Calculation:**

Given that

y=tanx, y=2sinx , -π3≤x≤π3 Since the area is symmetric about origin

A=2∫0π32sinx-tanxdx

By solving integral,

A=-4cosx+2lncosx0π3

By substituting limits,

=-4cosπ3+2lncosπ3--4cos0+2lncos0=-2+2ln1/2--4+2ln1

But ln1/2 = ln(2)-1 = -ln2 Thus by simplifying,

A=2-2ln(2)

Therefore,

A=2-2ln(2)

**Conclusion:**

The sketch of the region:

Area=2-2ln(2)