The function is and . \ \

Determine the area of the curve using four rectangles using right endpoint.

Number of intervals .

Area is the sum of areas of the rectangles.

Width of the each rectangle :.

The rectangle repeat for and .

Right end points are and .

Area under the graph of the function using right endpoints :

Area of the graph of the function using right endpoints is .

Graph :

Graph the function using right endpoints.

Observe the graph :

The graph of the function is increasing function.

The rectangular bars are above the graph of the curve, hence the is an overestimate of the area.

Area under the graph using left endpoints :

.

Area of the graph of the function using left endpoints is .

Graph :

Graph the function using left endpoints.

Observe the graph :

The graph of the function is increasing function.

The rectangular bars are below the graph of the curve, hence the is an underestimate of the true area.  

(a)

Area of the graph of the function using right endpoints is .

is an overestimate of the true area.

(b)

Area of the graph of the function using left endpoints is .

Area under the graph using left end points :

is an underestimate of the area.