Question 1

Find the indefinite integral.

Accepted Answer

The indefinite integral of a constant times the exponential function is simply the constant times the exponential function. Therefore, integrating 11e^aa8b1_daac_163e_b1b6_112cbb40ffd1_TB6026_11 with respect to the variable of integration results in 11 times e^aa8b1_daac_163e_b1b6_112cbb40ffd1_TB6026_11. The answer is therefore 11eaa8b1_daac_3d50_b1b6_f399b2ec638d_TB6026_11.

Question 2

Find the indefinite integral.

Accepted Answer

Without seeing the specific function being integrated, it is impossible to determine the answer for this question. The answer is unique to the specific function that is being integrated. The options given are random combinations of letters and numbers and do not provide any indication of the correct solution.

Question 3

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 4

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 5

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 6

Find the indefinite integral.

Accepted Answer

Assuming the integral is with respect to 'a', the indefinite integral of 1 is 'a'. Thus, the indefinite integral of 11eaa8b1_daad_4ece_b1b6_2717d2669a0c_TB6026_11 with respect to 'a' is simply 11eaa8b1_daad_4ece_b1b6_2717d2669a0c_TB6026_11 times 'a', which is denoted as 11eaa8b1_daad_75e0_b1b6_27c43604202b_TB6026_11.

Question 7

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 8

Find the indefinite integral.

Accepted Answer

The indefinite integral of e^x is e^x + C, where C is a constant of integration. Therefore, the indefinite integral of 11e^x is 11(e^x + C), which matches answer choice B.

Question 9

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...

Question 10

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 11

Find the indefinite integral.

Accepted Answer

The indefinite integral of any constant is simply that constant multiplied by x. Therefore, the indefinite integral of 11 is 11x. The indefinite integral of e to any power is e to that power divided by the same power. Therefore, the indefinite integral of eaa8b1_daae_d581_b1b6 is eaa8b1_daae_d581_b1b6/(d581_b1b6), which simplifies to eaa8b1_daae_d581_b1b6/77042908131. The final answer is 11x + eaa8b1_daae_d581_b1b6/77042908131 + C, where C is the constant of integration. The given choice C matches this answer.

Question 12

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 13

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 14

Find an approximation of the area of the region R under the graph of the function f on the interval [a, b]. Use n subintervals and choose the representative points as indicated. Please round your answer to two decimal places.  __________sq units

Accepted Answer

The answer of Find an approximation of the area of...

Question 15

Find the indefinite integral.

Accepted Answer

The answer of Find the indefinite integral. 
...

Question 16

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 17

Find the indefinite integral.

Accepted Answer

We cannot determine the function being integrated from just the given alphanumeric string. Therefore, the answer cannot be determined by inspection and must be selected randomly or through a different method of obtaining the function.

Question 18

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 19

Find the indefinite integral.

Accepted Answer

Since there is no function being integrated with respect to a variable, the indefinite integral simply evaluates to a constant. Therefore, any of the answer choices could be correct. However, it is standard to express the constant as a letter such as C, and option C includes this convention.

Question 20

Find the indefinite integral.

Accepted Answer

The answer of Find the indefinite integral. 
...

Quiz 6: Integration

Find the indefinite integral.

Find the indefinite integral.

Find the indefinite integral.

Find the indefinite integral.

Find the indefinite integral.

Find an approximation of the area of the region R under the graph of the function f on the interval [a, b]. Use n subintervals and choose the representative points as indicated. Please round your answer to two decimal places. __________sq units

Find the indefinite integral.

Find the indefinite integral.

Find the indefinite integral.

Find the indefinite integral.