Use a graphing utility to graph the region bounded by the graphs of the equations

Question

(a) Use a graphing utility to graph the region bounded by the graphs of the equations, (b) find the area of the region, and (c) Use the integration capabilities of the graphing utility to verify your results.

Answer 1

(a)

Step 1 :

The graphs of the equations are .

Sketch the region bounded by the graphs :

Graph the functions and .

Shade the region bounded by the curves between and .

Observe the graph for intersection points are and .

Solution:

Regions bounded by the graphs of equations is

 

Answer 2

(b)

Step 1 :

The graphs of the equations are .

Definite integral as area of the region:

If and are continuous and non-negative on the closed interval ,

then the area of the region bounded by the graphs of and and the vertical lines and is given by

.

The area of the region bounded by the curves contains 3 sub regions as shown below.

Region R₁ :

Upper curve : .

Lower curve : .

Region R₂ :

Upper curve : .

Lower curve : .

Region R₃ :

Upper curve : .

Lower curve : .

Comments

Contd....

Step 2 :

 Area bounded by the curves :

 The area bounded by the region is square units.

Solution:

The area bounded by the region is square units.

Answer 3

(c)

Step 1 :

The graphs of the equations are .

Verify the area region graphically :

Graphically the area bounded by region is 7.999 square units.

Solution:

The area bounded by the region is square units.