areas and distances

Question

. Estimate the area under the graph of f(x) = (1/ 1+4x^2) from x = −2 to x = 0 using four approximating rectangles and

(a) right endpoints (R4).

(b) left endpoints (L4).

(c) midpoints (M4). Carry any intermediate calculations to four decimal places, and round your final results to two decimal place

Answer 1

(a)

Step 1:

The function is  on interval is .

Number of rectangles are .

The sum of all circumscribed rectangle is upper sum.

, where .

Where .

Find the upper sum.

The right end point is .

Area of upper sum is .

Area of upper sum is  sq-units.

Solution:

Area of upper sum is  sq-units.

Comments

Contd...

Check the solution by using the graph.

Answer 2

(b)

Step 1:

The function is  on interval is .

Number of rectangles are .

The sum of all inscribed rectangle is upper sum.

, where .

Width  .

Find the lower sum.

The left end point is .

.

Area of lower sum is .

Area of lower sum is  sq-units.

Solution:

Area of lower sum is sq-units.

Comments

Contd...

Check the solution by using the graph.

Answer 3

(c)

Step 1:

The function is  on interval is .

Number of rectangles are .

Using Midpoint Rule:

The area is .

Consider .

.

Substitute in .

.

Substitute  values.

and .

.

Step 2:

Using Midpoint Rule:

Area =

Area of the region by using Midpoint Rule is sq-units.

Solution:

sq-units.

Comments

Contd...

Check the solution by using the graph.

.