Question 1

Let  and compute the Riemann sum of f over the interval [1, 2] , using
Two subintervals of equal length (n = 2).Choose the representative points to be the midpoints of the subintervals.
__________ sq units
Five subintervals of equal length (n = 5).Choose the representative points to be the midpoints of the subintervals.
__________ sq units
Ten subintervals of equal length (n = 10).Choose the representative points to be the midpoints of the subintervals.
__________ sq units

Accepted Answer

The answer of Let  and compute the Riemann sum...

Question 2

Let  and compute the Riemann sum of f over the interval  , choosing the representative points to be the left endpoints of the subintervals, using:
Two subintervals of equal length (  ).
__________ sq units
Five subintervals of equal length (  ).
__________ sq units
Ten subintervals of equal length (  ).
__________ sq units
Can you guess at the area of the region under the graph of f on the interval  ?  __________ sq units

Accepted Answer

The answer of Let  and compute the Riemann sum...

Question 3

Let  and compute the Riemann sum of f over the interval [0, 1] , using
Two subintervals of equal length (n = 2).Choose the representative points to be the right end points of the subintervals.
__________ sq units
Five subintervals of equal length (n = 5).Choose the representative points to be the right end points of the subintervals.
__________ sq units
Ten subintervals of equal length (n = 10).Choose the representative points to be the right end points of the subintervals.
__________ sq units

Accepted Answer

The answer of Let  and compute the Riemann sum...

Question 4

Find the indefinite integral.

Accepted Answer

The answer of Find the indefinite integral. 
...

Question 5

Find the indefinite integral.

Accepted Answer

The answer of Find the indefinite integral. 
...

Question 6

Find the indefinite integral.

Accepted Answer

The answer of Find the indefinite integral. 
...

Question 7

Figure (a) shows a vacant lot with a 80-ft frontage in a development.To estimate its area, we introduce a coordinate system so that the x-axis coincides with the edge of the straight road forming the lower boundary of the property, as shown in Figure (b).Then, thinking of the upper boundary of the property as the graph of a continuous function f over the interval [0, 80], we see that the problem is mathematically equivalent to that of finding the area under the graph of f on [0, 80].To estimate the area of the lot using a Riemann sum, we divide the interval [0, 80] into four equal subintervals of length 20 ft.Then, using surveyor's equipment, we measure the distance from the midpoint of each of these subintervals to the upper boundary of the property.These measurements give the values of f(x) at x = 10, 30, 50, and 70.What is the approximate area of the lot?   __________ square feet

Accepted Answer

The answer of Figure (a) shows a vacant lot with...

Question 8

Let f(x) = x
Use a Riemann sum with four subintervals of equal length (n = 4) to approximate the area of R (under the graph of f on the interval [0, 2]).Choose the representative points to be the left end points of the subintervals.
__________ sq units
Repeat previous part with eight subintervals of equal length (n = 8).
__________ sq units
Compare the approximations obtained in previous parts with the exact area (2 sq units).Do the approximations improve with larger n?

Accepted Answer

The answer of Let f(x) = x
Use a Riemann sum...

Question 9

Let  and compute the Riemann sum of f over the interval  , choosing the representative points to be the left endpoints of the subintervals, using:
Two subintervals of equal length (  ).
__________ sq units
Five subintervals of equal length (  ).
__________ sq units
Ten subintervals of equal length (  ).
__________ sq units
Can you guess at the area of the region under the graph of f on the interval  ?  __________ sq units

Accepted Answer

The answer of Let  and compute the Riemann sum...

Question 10

Find the indefinite integral.

Accepted Answer

The answer of Find the indefinite integral. 
...

Question 11

Let  and compute the Riemann sum of f over the interval  , choosing the representative points to be the midpoints of the subintervals, using:
Two subintervals of equal length (  ).
__________ sq units
Five subintervals of equal length (  ).
__________ sq units
Ten subintervals of equal length (  ).
__________ sq units
Can you guess at the area of the region under the graph of f on the interval  ?  __________sq units

Accepted Answer

The answer of Let  and compute the Riemann sum...

Question 12

Let f(x) = 8 - 4x
Use a Riemann sum with four subintervals of equal length (n = 5) to approximate the area of R (under the graph of f on the interval [0, 2]).Choose the representative points to be the left end points of the subintervals.
__________ sq units
Repeat previous part with eight subintervals of equal length (n = 10).
__________ sq units
Compare the approximations obtained in previous parts with the exact area (8 sq units).Do the approximations improve with larger n?

Accepted Answer

The answer of Let f(x) = 8 - 4x
Use a...

Question 13

Find the indefinite integral.

Accepted Answer

The answer of Find the indefinite integral. 
...

Question 14

Find the indefinite integral.

Accepted Answer

The answer of Find the indefinite integral. 
...

Question 15

Find the indefinite integral.

Accepted Answer

The answer of Find the indefinite integral. 
...

Question 16

Find the indefinite integral.

Accepted Answer

The answer of Find the indefinite integral. 
...

Question 17

Find the indefinite integral.

Accepted Answer

The answer of Find the indefinite integral. 
...

Question 18

Let  and compute the Riemann sum of f over the interval [2, 4] , using
Two subintervals of equal length (n = 2).Choose the representative points to be the right end points of the subintervals.
__________ sq units
Five subintervals of equal length (n = 5).Choose the representative points to be the right end points of the subintervals.
__________ sq units
Ten subintervals of equal length (n = 10).Choose the representative points to be the right end points of the subintervals.
__________ sq units

Accepted Answer

The answer of Let  and compute the Riemann sum...

Question 19

Find the indefinite integral.

Accepted Answer

The answer of Find the indefinite integral. 
...

Question 20

Figure (a) shows a vacant lot with a 100-ft frontage in a development.To estimate its area, we introduce a coordinate system so that the x-axis coincides with the edge of the straight road forming the lower boundary of the property, as shown in Figure (b).Then, thinking of the upper boundary of the property as the graph of a continuous function f over the interval [0, 100], we see that the problem is mathematically equivalent to that of finding the area under the graph of f on [0, 100].To estimate the area of the lot using a Riemann sum, we divide the interval [0, 100] into five equal subintervals of length 20 ft.Then, using surveyor's equipment, we measure the distance from the midpoint of each of these subintervals to the upper boundary of the property.These measurements give the values of f(x) at x = 10, 30, 50, 70, and 90.What is the approximate area of the lot?   __________ square feet

Accepted Answer

The answer of Figure (a) shows a vacant lot with...

Quiz 6: Integration

Find the indefinite integral.

Find the indefinite integral.

Find the indefinite integral.

Find the indefinite integral.

Find the indefinite integral.

Find the indefinite integral.

Find the indefinite integral.

Find the indefinite integral.

Find the indefinite integral.

Find the indefinite integral.