Question

6.3 3a.  Find an approximation of the area of the region R under the graph of the function f on the interval [−1, 2]. Use n = 6 subintervals. Choose the representative points to be the left endpoints of the subintervals.  f(x) = 6 − x²   

_________square units

b. Find an approximation of the area of the region R under the graph of the function f on the interval [1, 3]. Use n = 4 subintervals. Choose the representative points to be the right endpoints of the subintervals.  f(x) = 6/x

_________ square units

c. Find an approximation of the area of the region R under the graph of the function f on the interval [0, 3]. Use n = 5 subintervals. Choose the representative points to be the midpoints of the subintervals. (Round your answer to one decimal place.) f(x) = 3e^x

___________square units

Answer 1

6.3 , 3a)

To approximate the area of the region under the graph of on the interval [-1,2] by using 6 subintervals.

Where,

Given, a= -1 , b= 2 and n = 6

Thus,

Now , divide the interval [-1,2] into 6 subinterval with width 1/2.

Now, find the value of the function at left end points

,

,

Now, plug all values in the formula

Thus, Therefore, the area of the region is 15.625 square unit.

Part b )

To approximate the area of the region under the graph of on the interval [1,3] by using n=4 subintervals.

Where,

Given, a= 1 , b= 3 and n =4

Thus,

Now , divide the interval [1,3] into 4 subinterval with width 1/2.

Now, find the value of the function at Right end points

,

Now, plug all values in the formula

Thus, Therefore, the area of the region is 5.7 square unit.

Part c)

To approximate the area of the region under the graph of on the interval [0,3] by using 5 subintervals by using midpoint formula.

Where, and are midpoints of the subintervals.

Given, a= 0 , b= 3 and n = 5

Thus,

Now , divide the interval [0,3] into 5 subinterval with width 3/5.

To find midpoints of each interval use the mid point formula, midpoint of the inter [a,b] is

Thus, midpoints of each interval are

Now, find the value of the function at each midpoints,

,

Now, plug all values in the formula

Thus, Therefore, the area of the region is 56.4067square unit.