find the area of the region bounded by the graph of the function

Question

1. (a) Find the area of the region bounded by the graph of the function f(x) = x^2 + 3, the X-axis and the lines x = −1 to x = 2.

    (b) Find the area of the region bounded by the graph of the function f(x) = e^x + 2, the X-axis and the lines x = 0 to x = 3.

 

 2. (a) Find the area of the region enclosed by functions f(x) = x^2 and g(x) = x

     (b) Find the area of the region bounded by the graph of the function f(x) = x^2 − 9 and g(x) = 9 − x^2 .

 

3. Find the area of the region bounded by the graph of the function f(x) = x^2−4, the X-axis and the lines x = −4 to x = 3.

Answer 1

1(a)

Step 1:

The curves are and -axis and the lines are , .

Area of the region between two curves:

If and  are continuous on and for all in , then the area of the region bounded by the graphs of and and the vertical lines and is .

Here, , , and .

Graph:

Graph the functions , and the vertical lines and .

Observe the graph:

The two functions on the interval .

Step 2:

Find the area of the region.

Substitute , and in .

Apply power rule of integration: .

.

Area bounded by the two curves is .

Solution:

Area bounded by the two curves is .

Answer 2

1(b)

Step 1:

The curves are , -axis and the lines are , .

The area of the region bounded by the graphs of and and the vertical lines and is .

Here, , , and .

Graph:

Graph the functions , and the vertical lines and .

Observe the graph:

The two functions on the interval .

Step 2:

Find the area of the region.

Substitute , and in .

Basic integral formula: .

.

Area bounded by the two curves is .

Solution:

Area bounded by the two curves is .

Answer 3

2(a)

Step 1:

The functions are  and .

The area of the region bounded by the graphs of and and the vertical lines and is .

Graph:

Graph the functions and .

Observe the graph:

The intersection points of the two curves are and .

in the interval .

Step 2:

Find the area of the region.

Substitute , and in .

Apply power rule of integration: .

.

Area bounded by the functions is .

Solution:

Area bounded by the functions is .

Answer 4

2(b)

Step 1:

The curve equations are and  .

Graph the functions  and .

Shade the required region.

Observe the graph :

Region enclosed by the curves is between and .

Step 2:

Area between the curves and and between and is 

.

Here and .

Area of the region: .

Integral of symmetric function: If is even function, then .

Area of the region is sq-units.

Solution:

Area of the region is sq-units.

Answer 5

(3)

Step 1:

The curve equations are , , the lines are and .

Graph the functions  , ,   and .

Shade the required region.

Observe the graph :

Region enclosed by the curves is between and .

Step 2:

Area between the curves and and between and is 

.

Observe the graph:

In the interval ,

Upper curve is and the lower curve is .

In the interval ,

Upper curve is and the lower curve is .

In the interval ,

Upper curve is and the lower curve is .

Area of the region: .

Area of the region is sq-units.

Solution:

Area of the region is sq-units.