Try It Yourself#

Exercise 1#

Using a right Riemann sum with 3 subintervals of equal length, approximate

\[\int_2^{8} \frac{1}{x}~dx.\]
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Answer: \(13/12\)

Exercise 2#

Using a left Riemann sum with 3 subintervals of equal length, approximate

\[\int_2^9 3x - 4 ~dx.\]
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Answer: \(63\)

Exercise 3#

Evaluate

\[\displaystyle \int_0^2 e^{3x} + \frac{x^2}{3}+5x ~dx.\]
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Answer: \(e^6/3 + 95/9\)

Exercise 4#

Find the area under the graph of

\[f(x) = \dfrac{\ln(x)}{x}\]

over the interval \([e^4, e^7].\)

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Answer: \(33/2\)

Exercise 5#

The marginal cost function associated with producing \(x\) widgets is given by

\[C'(x) = -0.4x + 75\]

where \(C'(x)\) is measured in dollars/unit and \(x\) denotes the number of widgets. If the daily fixed costs incurred in the production amount to \(\$600\), find the total cost incurred in producing the first 100 units of the day.

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Answer: \(\$6,100\)

Exercise 6#

Evaluate

\[\displaystyle \int_0^4 \frac{x}{2x^2+10}~dx.\]
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Answer: \(\frac{1}{4}\ln(\frac{21}{5})\)

Exercise 7#

Evaluate

\[\int_0^{\ln(3)/3} \frac{e^{3x}}{\left(e^{3x}+1\right)^3}~dx.\]
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Answer: \(1/32\)

Exercise 8#

Compute the average value of

\[f(x) = x^2+8x-2\]

on the interval \([-2, 2]\).

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Answer: \(-2/3\)