#### To determine

**To find:**

The area of the triangle with the given vertices using calculus.

#### Answer

i) The triangle with the given vertices.

ii) The area of the enclosed region:

A=2

#### 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)

Equation of line passing through two points (a,b) and (c,d) is given by y-bx-a=d-bc-a

**2) Given:**

The vertices 2, 0, 0,2, (-1,1)

3) **Calculation:**

First, find the equations of lines passing through the given vertices.

An equation of the line passing through (2, 0) and (0, 2) is

y=-x+2

An equation of the line passing through (2, 0) and (-1,1) is

y=-13x+23

And the equation of the line passing through (0, 2) and -1,1 is

y=x+2

Now, use these equations to find the area enclosed by the given vertices.

Similar to Excercise 33, therefore,

A=∫-10(x+2)--13x+23dx+∫02-x+2--13x+23 dx

=∫-1043x+43dx+∫02-23x+43 dx

Integrating this gives,

A=23x2+43x0-1+-13x2+43x20

Plugging the values

A=0-23-43+-43+83-0

A=0+23+43

A=63

A=2

**Conclusion:**

i) The triangle enclosed by the given vertices

ii) Area of the shaded region:

A=2