Mathematics for Management -- Supplementary Electronic Materials

Quiz: Definite Integrals

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1. Use a Riemann sum (left endpoints) with only one sub-interval to estimate $ \int ^ 1_0 \left( x + 1\right) \textrm{d} x $

$ 1 $ $ 2 $ $ 0 $ $ 3 $

2. Based on the table, use a Riemann sum (left endpoints) and 4 sub-intervals to estimate the area under the curve. (Choose the correct set-up.) $$ \begin{array}{c || c c c c c} x & 0 & 5 & 6 & 8 & 9 \\ \hline f(x) & 3 & 4 & 5 & 7 & 6 \end{array} $$

$ 5 \cdot 2 + 1 \cdot 4 + 2 \cdot 5 + 1 \cdot 7 $ $ 5 \cdot 3 + 6 \cdot 4 + 8 \cdot 5 + 9 \cdot 7 $ $ 5 \cdot 4 + 1 \cdot 5 + 2 \cdot 7 + 1 \cdot 6 $ $ 0 \cdot 3 + 5 \cdot 4 + 6 \cdot 5 + 8 \cdot 7 $

3. Based on the table, determine the value of the Riemann sum (left endpoints) with 3 sub-intervals to estimate the area under the curve. $$ \begin{array}{c || c c c c} x & 1 & 4 & 7 & 8 \\ \hline f(x) & 2 & 4 & 5 & 7 \end{array} $$

$ 28 $ $ 34 $ $ 16 $ $ 23 $

4. Based on the table, determine the value of the Riemann sum (left endpoints) with 4 equally spaced sub-intervals to estimate the area under the curve. $$ \begin{array}{c || c c c c c} x & 0 & 1 & 2 & 3 & 4 \\ \hline f(x) & 4 & 8 & 5 & 3 & 7 \end{array} $$

$ 20 $ $ 23 $ $ 26 $ $ 29 $

5. Based on the table, determine the value of the Riemann sum (left endpoints) with 4 sub-intervals to estimate the area under the curve. $$ \begin{array}{c || c c c c c} x & 2 & 3 & 5 & 8 & 13 \\ \hline f(x) & 6 & -2 & -1 & 3 & 9 \end{array} $$

$ 6 $ $ 14 $ $ 28 $ $ 32 $

6. Suppose a car's velocity was measured every second and the results are recorded in the table below: $$ \begin{array}{l || c c c c c c c c c c c c} \text{time [sec]} & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ \hline \text{velocity [cm/sec]} & 20 & 26 & 30 & 35 & 38 & 42 & 44 & 46 & 48 & 49 & 50 \end{array} $$ Determine the distance traveled using a Riemann sum (left endpoints) with 5 sub-intervals.

$ 360 $ $ 396 $ $ 420 $ $ 390 $

7. Estimate the value of the definite integral of $ f(x) = 3x^2 + 1 $ from 0 to 3 using a Riemann sum (left endpoints) with 6 equally spaced sub-intervals.

$ 37.125 $ $ 74.25 $ $ 23.625 $ $ 23.125 $

8. Estimate the value of the definite integral of $ f(x) = x^4 + 1 $ from -2 to 2 using a Riemann sum (left endpoints) with 4 equally spaced sub-intervals.

$ -12 $ $ 10 $ $ 15 $ $ 22 $

9. se a Riemann sum (left endpoints) with 3 equally spaced sub-intervals to estimate $ \int^{3 \pi/2}_0 \cos(x) \, \textrm{d} x \, . $

$ -1 $ $ 0 $ $ 1 $ $ \pi $

10. The following expression is a Riemann sum approximation for which definite integral? $$ \frac{1}{50} \left( \sqrt{\frac{1}{50}} + \sqrt{\frac{2}{50}} + \sqrt{\frac{3}{50}} + \dots + \sqrt{\frac{49}{50}}\right) \, . $$

$ \int^1_0 \sqrt{\frac{x}{50}} \textrm{d} x $ $ \int^1_0 \sqrt{x} \textrm{d} x $ $ \frac{1}{50} \int^1_0 \sqrt{\frac{x}{50}} \textrm{d} x $ $ \frac{1}{50} \int^1_0 \sqrt{x} \textrm{d} x $

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Solution: 1a; 2a; 3d; 4a; 5b; 6a; 7c; 8d; 9b; 10b

Here, a, b, c, d indicate the 1st, 2nd, 3rd, and 4th answer choice, respectively, for the numbered questions.

Quiz: Definite Integrals

1. Use a Riemann sum (left endpoints) with only one sub-interval to estimate \( \int ^ 1_0 \left( x + 1\right) \textrm{d} x \)

2. Based on the table, use a Riemann sum (left endpoints) and 4 sub-intervals to estimate the area under the curve. (Choose the correct set-up.) $$ \begin{array}{c || c c c c c} x & 0 & 5 & 6 & 8 & 9 \\ \hline f(x) & 3 & 4 & 5 & 7 & 6 \end{array} $$

3. Based on the table, determine the value of the Riemann sum (left endpoints) with 3 sub-intervals to estimate the area under the curve. $$ \begin{array}{c || c c c c} x & 1 & 4 & 7 & 8 \\ \hline f(x) & 2 & 4 & 5 & 7 \end{array} $$

4. Based on the table, determine the value of the Riemann sum (left endpoints) with 4 equally spaced sub-intervals to estimate the area under the curve. $$ \begin{array}{c || c c c c c} x & 0 & 1 & 2 & 3 & 4 \\ \hline f(x) & 4 & 8 & 5 & 3 & 7 \end{array} $$

5. Based on the table, determine the value of the Riemann sum (left endpoints) with 4 sub-intervals to estimate the area under the curve. $$ \begin{array}{c || c c c c c} x & 2 & 3 & 5 & 8 & 13 \\ \hline f(x) & 6 & -2 & -1 & 3 & 9 \end{array} $$

7. Estimate the value of the definite integral of \( f(x) = 3x^2 + 1 \) from 0 to 3 using a Riemann sum (left endpoints) with 6 equally spaced sub-intervals.

8. Estimate the value of the definite integral of \( f(x) = x^4 + 1 \) from -2 to 2 using a Riemann sum (left endpoints) with 4 equally spaced sub-intervals.

9. se a Riemann sum (left endpoints) with 3 equally spaced sub-intervals to estimate \( \int^{3 \pi/2}_0 \cos(x) \, \textrm{d} x \, . \)

10. The following expression is a Riemann sum approximation for which definite integral? $$ \frac{1}{50} \left( \sqrt{\frac{1}{50}} + \sqrt{\frac{2}{50}} + \sqrt{\frac{3}{50}} + \dots + \sqrt{\frac{49}{50}}\right) \, . $$