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) explain why  the area of the region is difficult to find by hand, and (c) use the integration capabilities of the graphing utility to approximate the area to four decimal places.

Answer 1

(a)

Step 1 :

The equations are and .

Sketch the region bounded curves :

Graph the functions and .

Shade the region bounded by the curves between and .

Note : Here the intersection points are found by using graphing utility.

Observe the graph for intersection points are and .

Solution:

Region bounded by the curves is

 

Answer 2

(b)

Step 1 :

The equations are and .

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

.

So, the area of region is .

Point of intersection :

The equations are and .

The intersections points can be found by equating the above two equations.

It is difficult to find the intersection for cosine function and a curve equation.

Therefore, it is difficult to evaluate the integral without using graphing utility.

Solution:

Area of the region is difficult to find by hand.

Answer 3

(c)

Step 1 :

The equations are and .

Verify the area region graphically :

Graphically the area bounded by shaded region is 6.297 square units.

Solution:

The area bounded by the shaded region is square units.