Question 1

Use the graph of f shown in the figure to evaluate the integral by interpreting it geometrically.

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Question 2

Given that  , find  .

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Question 3

Find an expression for the area under the graph of  as a limit. Do not evaluate the limit.

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Question 4

Use the Midpoint Rule with n = 10 to approximate the integral.

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Question 5

Evaluate the integral by interpreting it in terms of areas.

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Question 6

Evaluate the limit by interpreting it as the limit of a Riemann sum of a function on the interval [a, b].

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Question 7

If  , find the Riemann sum with n = 5 correct to 3 decimal places, taking the sample points to be midpoints.

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Question 8

Use the Midpoint Rule with  to approximate the integral. Round the answer to 3 decimal places.

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Question 9

The given expression is the limit of a Riemann sum of a function f on [a, b]. Write this expression as a definite integral on [a, b].

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Question 10

Evaluate  by interpreting it in terms of areas.

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Question 11

Find an expression for the area under the graph of f as a limit. Do not evaluate the limit.  ,

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Question 12

Express the sum as a single integral in the form  .

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Question 13

Use  to evaluate the integral.

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...

Question 14

If  and  , find  .

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Question 15

Evaluate the Riemann sum for  ,  with four subintervals, taking the sample points to be right endpoints.

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Question 16

A table of values of an increasing function  is shown. Use the table to find an upper estimate of  .    -49  -38  -23  2  13  31

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Question 17

Evaluate the Riemann sum for  with four subintervals, taking the sample points to be right endpoints.

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Question 18

The table gives the values of a function obtained from an experiment. Use the values to estimate  using three equal subintervals with left endpoints.
w
0
1
2
3
4
5
6
f (w)
9.7
9.1
7.7
6.1
4.2
-6.6
-10.3

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The answer of The table gives the values of a...

Question 19

Use the Midpoint Rule with n = 5 to approximate the integral.  Round your answer to three decimal places.

Accepted Answer

Using the Midpoint Rule with n = 5 gives:
Δx = (2-0)/5 = 0.4
x1 = 0.2, x2 = 0.6, x3 = 1, x4 = 1.4, x5 = 1.8
f(x1) = e^(0.2) ≈ 1.221
f(x2) = e^(0.6) ≈ 1.822
f(x3) = e^(1) ≈ 2.718
f(x4) = e^(1.4) ≈ 4.055
f(x5) = e^(1.8) ≈ 6.049
Using the Midpoint Rule formula:
∫₀² e^x dx ≈ Δx [f(x1) + f(x2) + f(x3) + f(x4) + f(x5)]
≈ 0.4 [1.221 + 1.822 + 2.718 + 4.055 + 6.049]
≈ 5.693
Rounded to three decimal places, the answer is 5.693, which corresponds to choice C.

Question 20

Use the given graph of  to find the Riemann sum with six subintervals. Take the sample points to be left endpoints.

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Quiz 4: Integrals

Use the graph of f shown in the figure to evaluate the integral by interpreting it geometrically.

Given that , find .

Find an expression for the area under the graph of as a limit. Do not evaluate the limit.

Use the Midpoint Rule with n = 10 to approximate the integral.

Evaluate the integral by interpreting it in terms of areas.

Evaluate the limit by interpreting it as the limit of a Riemann sum of a function on the interval [a, b].

If , find the Riemann sum with n = 5 correct to 3 decimal places, taking the sample points to be midpoints.

Use the Midpoint Rule with to approximate the integral. Round the answer to 3 decimal places.

The given expression is the limit of a Riemann sum of a function f on [a, b]. Write this expression as a definite integral on [a, b].

Evaluate by interpreting it in terms of areas.

Find an expression for the area under the graph of f as a limit. Do not evaluate the limit. ,

Express the sum as a single integral in the form .

Use to evaluate the integral.

If and , find .

Evaluate the Riemann sum for , with four subintervals, taking the sample points to be right endpoints.

A table of values of an increasing function is shown. Use the table to find an upper estimate of . -49 -38 -23 2 13 31

Evaluate the Riemann sum for with four subintervals, taking the sample points to be right endpoints.

The table gives the values of a function obtained from an experiment. Use the values to estimate using three equal subintervals with left endpoints. w 0 1 2 3 4 5 6 f (w) 9.7 9.1 7.7 6.1 4.2 -6.6 -10.3

Use the Midpoint Rule with n = 5 to approximate the integral. Round your answer to three decimal places.

Use the given graph of to find the Riemann sum with six subintervals. Take the sample points to be left endpoints.