Try It Yourself#
Exercise 1#
Riemann sums
Using a right Riemann sum with 3 subintervals of equal length, approximate \(\displaystyle \int_2^{8} \frac{1}{x}~dx\).
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Answer: \(13/12\)
Exercise 2#
Riemann sums
Using a left Riemann sum with 3 subintervals of equal length, approximate \(\displaystyle \int_2^9 3x - 4 ~dx\).
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Answer: \(63\)
Exercise 3#
Definite integral
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#
Area of a region
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#
Total cost
The marginal cost function associated with producing \(x\) widgets is given by
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#
Definite integral
Evaluate \(\displaystyle \int_0^4 \frac{x}{2x^2+10}~dx\).
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Answer: \(\dfrac{1}{4}\ln(21/5)\)
Exercise 7#
Definite integral
Evaluate \(\displaystyle \int_0^{\ln(3)/3} \frac{e^{3x}}{\left(e^{3x}+1\right)^3}~dx\).
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Answer: \(1/32\)
Exercise 8#
Average value
Compute the average value of \(f(x) = x^2+8x-2\) on the interval \([-2, 2]\).
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Answer: \(-2/3\)