Question 1

By reading values from the given graph of f , use five rectangles to find a lower estimate, to the nearest whole number, for the area from 0 to 10 under the given graph of f .

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Question 2

Evaluate the limit after first finding the sum (as a function of n) using the summation formulas.

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Question 3

Estimate the area from 0 to 5 under the graph of  using five approximating rectangles and right endpoints.

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Question 4

The velocity graph of a braking car is shown. Use it to estimate to the nearest foot the distance traveled by the car while the brakes are applied.
Use a left sum with n = 7.

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Question 5

You are given function f defined on an interval [a, b], the number n of subintervals of equal length , and the evaluation points c_k in . (a) Sketch the graph of f and the rectangles associated with the Riemann sum for f on [a, b], and (b) find the Riemann sum. , c_k is the right endpoint

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Question 6

Express the limit as a definite integral on the given interval.

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Question 7

Approximate the area under the curve  from  using  approximating rectangles of equal widths and right endpoints. The choices are rounded to the nearest hundredth.

Accepted Answer

Using 6 rectangles of equal width, we have:

$$\Delta x = \frac{6.4 - 1}{6} \approx 1.07$$

The right endpoints of the rectangles are: $$1.07, 2.14, 3.21, 4.28, 5.35, 6.42$$

The area of each rectangle is:

$$\Delta A_i \approx f(x_i) \Delta x$$

where $x_i$ is the right endpoint of the $i$th rectangle.

Thus, we have:

$$\begin{aligned} \Delta A_1 &\approx f(1.07) 	imes 1.07 \approx 11.06 \ \Delta A_2 &\approx f(2.14) 	imes 1.07 \approx 25.94 \ \Delta A_3 &\approx f(3.21) 	imes 1.07 \approx 51.34 \ \Delta A_4 &\approx f(4.28) 	imes 1.07 \approx 87.35 \ \Delta A_5 &\approx f(5.35) 	imes 1.07 \approx 132.97 \ \Delta A_6 &\approx f(6.42) 	imes 1.07 \approx 187.20 \end{aligned}$$

Summing up the areas of the rectangles gives an approximation of the integral:

$$\begin{aligned} A &\approx \Delta A_1 + \Delta A_2 + \Delta A_3 + \Delta A_4 + \Delta A_5 + \Delta A_6 \ &\approx 11.06 + 25.94 + 51.34 + 87.35 + 132.97 + 187.20 \ &\approx 495.86 \end{aligned}$$

Therefore, the closest choice is C) 495.60.

Question 8

Approximate the area under the curve  from 1 to 2 using ten approximating rectangles of equal widths and right endpoints. Round the answer to the nearest hundredth.

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Question 9

Evaluate by interpreting it in terms of areas.

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Question 10

The speed of a runner increased steadily during the first three seconds of a race. Her speed at half-second intervals is given in the table. Find a lower estimate for the distance that she traveled during these three seconds.           2.5
4.4
6.5
9.4
12.2
16.8

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Question 11

Evaluate  .

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Question 12

Determine a region whose area is equal to  .

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Question 13

Evaluate by interpreting it in terms of areas.

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Question 14

The graph of a function f on the interval [0, 8] is shown in the figure. Compute the Riemann sum for f on [0, 8] using four subintervals of equal length and choosing the evaluation points to be (a) the left endpoints, (b) the right endpoints, and (c) the midpoints of the subintervals.

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Question 15

Find the limit.

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Quiz 4: Integrals

By reading values from the given graph of f , use five rectangles to find a lower estimate, to the nearest whole number, for the area from 0 to 10 under the given graph of f .

Evaluate the limit after first finding the sum (as a function of n) using the summation formulas.

Estimate the area from 0 to 5 under the graph of using five approximating rectangles and right endpoints.

The velocity graph of a braking car is shown. Use it to estimate to the nearest foot the distance traveled by the car while the brakes are applied. Use a left sum with n = 7.

You are given function f defined on an interval [a, b], the number n of subintervals of equal length , and the evaluation points ck in . (a) Sketch the graph of f and the rectangles associated with the Riemann sum for f on [a, b], and (b) find the Riemann sum. , ck is the right endpoint

Express the limit as a definite integral on the given interval.

Approximate the area under the curve from using approximating rectangles of equal widths and right endpoints. The choices are rounded to the nearest hundredth.

Approximate the area under the curve from 1 to 2 using ten approximating rectangles of equal widths and right endpoints. Round the answer to the nearest hundredth.

Evaluate by interpreting it in terms of areas.

The speed of a runner increased steadily during the first three seconds of a race. Her speed at half-second intervals is given in the table. Find a lower estimate for the distance that she traveled during these three seconds. 2.5 4.4 6.5 9.4 12.2 16.8

Evaluate .

Determine a region whose area is equal to .

Evaluate by interpreting it in terms of areas.

The graph of a function f on the interval [0, 8] is shown in the figure. Compute the Riemann sum for f on [0, 8] using four subintervals of equal length and choosing the evaluation points to be (a) the left endpoints, (b) the right endpoints, and (c) the midpoints of the subintervals.

Find the limit.

You are given function f defined on an interval [a, b], the number n of subintervals of equal length , and the evaluation points c_k in . (a) Sketch the graph of f and the rectangles associated with the Riemann sum for f on [a, b], and (b) find the Riemann sum. , c_k is the right endpoint