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3.7.E: Exercises

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    3.7 Exercises

    Exercise \(\PageIndex{1}-\PageIndex{4}\)

    In problems 1 – 4, use the values in the table to estimate the areas.

    \(x\) \(f(x)\) \(g(x)\) \(h(x)\)

    0

    5

    2

    5

    1

    6

    1

    6

    2

    6

    2

    8

    3

    4

    2

    6

    4

    3

    3

    5

    5

    2

    4

    4

    6

    2

    0

    2

    1. Estimate the area between \(f\) and \(g\), between \(x = 0\) and \(x = 4\).

    2. Estimate the area between \(g\) and \(h\), between \(x = 0\) and \(x = 6\).

    3. Estimate the area between \(f\) and \(h\), between \(x = 0\) and \(x = 4\).

    4. Estimate the area between \(f\) and \(g\), between \(x = 0\) and \(x = 6\).

    Exercise \(\PageIndex{1}\)

    Estimate the area of the island shown

    clipboard_ef149260bb44e87dc4a723e98773d48fc.png
    Exercise \(\PageIndex{6}-\PageIndex{15}\)

    In problems 6 – 15, find the area between the graphs of \(f\) and \(g\) for \(x\) in the given interval. Remember to draw the graph!

    6. \(f(x) = x^2 + 3 \), \(g(x) = 1\) and \(–1 \leq x \leq 2\).
    7. \(f(x) = x^2 + 3 \), \(g(x) = 1 + x\) and \(0 \leq x \leq 3\).
    8. \(f(x) = x^2 \), \(g(x) = x\) and \(0 \leq x \leq 2\).
    9. \(f(x) = (x –1)^2 \), \(g(x) = x + 1\) and \(0 \leq x \leq 3\).
    10. \(f(x) = \frac{1}{x}\), \(g(x) = x\) and \(1 \leq x \leq e\).
    11. \(f(x) = \sqrt{x}\), \(g(x) = x\) and \(0 \leq x \leq 4\).
    12. \(f(x) = 4 – x^2 \), \(g(x) = x + 2\) and \(0 \leq x \leq 2\).
    13. \(f(x) = e^x\), \(g(x) = x\) and \(0 \leq x \leq 2\).
    14. \(f(x) = 3 \), \(g(x) = \sqrt{1-x^2}\) and \(0 \leq x \leq 1\).
    15. \(f(x) = 2 \), \(g(x) = \sqrt{4-x^2}\) and \(–2 \leq x \leq 2\).
    Exercise \(\PageIndex{16}-\PageIndex{18}\)

    For problems 16-18, find the volume of the solid obtained by rotating the specified region about the \(x\) axis.

    16. Region under \(f(x) = x^2 + 3\) for \(–1 \leq x \leq 2\).

    17. Region under \(f(x) = 4 – x^2\) for \(0 \leq x \leq 2\).

    18. Region under \(f(x) = \frac{1}{x}\) for \(1 \leq x \leq 2\).

    Exercise \(\PageIndex{19}-\PageIndex{20}\)

    In problems 19 and 20 use the values in the table to estimate the average values.

    \(x\) \(f(x)\) \(g(x)\)

    0

    5

    2

    1

    6

    1

    2

    6

    2

    3

    4

    2

    4

    3

    3

    5

    2

    4

    6

    2

    0

    19. Estimate the average value of \(f\) on the interval [0, 6].

    20. Estimate the average value of \(g\) on the interval [0, 6].

    Exercise \(\PageIndex{21}-\PageIndex{26}\)

    In problems 21 – 26, find the average value of \(f\) on the given interval.

    clipboard_e92ed5e431e03d01df80201a7aabad2cf.png

    21. \(f(x)\) from the graph for \(0 \leq x \leq 2\).

    22. \(f(x)\) from the graph for \(0 \leq x \leq 4\).

    23. \(f(x)\) from the graph for \(1 \leq x \leq 6\).

    24. \(f(x)\) from the graph for \(4 \leq x \leq 6\).

    25. \(f(x) = 2x + 1\) for \(0 \leq x \leq 4\).

    26. \(f(x) = x^2\) for \(0 \leq x \leq 2\).

    Exercise \(\PageIndex{27}\)

    The graph shows the velocity of a car during a 5 hour trip.

    clipboard_e61f3359186c7acd43ea6760dcda3e8ed.png

    (a) Estimate how far the car traveled during the 5 hours.

    (b) At what constant velocity should you drive in order to travel the same distance in 5 hours?

    Exercise \(\PageIndex{28}\)

    The graph shows the number of telephone calls per minute at a large company.

    clipboard_ea06d76d797ded4f77e375886ae77d1f2.png

    Estimate the average number of calls per minute

    (a) From 8 am to 5 pm.

    (b) From 9 am to 1 pm.

    This page titled 3.7.E: Exercises was last modified on Tue, 09 Jul 2024 03:44:25 GMT and is shared under a CC BY 3.0 license and was authored, remixed, and/or curated by Shana Calaway, Dale Hoffman, and David Lippman (The OpenTextBookStore) via source content that was edited to the style and standards of the LibreTexts platform.