StudySmarter AI is coming soon!

- :00Days
- :00Hours
- :00Mins
- 00Seconds

A new era for learning is coming soonSign up for free

Suggested languages for you:

Americas

Europe

Q15E

Expert-verifiedFound in: Page 298

Book edition
2nd

Author(s)
James Stewart

Pages
830 pages

ISBN
9781133112280

**Find the average value of the function \(g(x) = \sqrt(3){x}\) in the interval \({\rm{(1,8)}}\).**

The average value of the function \(g(x) = \sqrt(3){x}\) in the interval \({\rm{(1,8)}}\) is \(\frac{{45}}{{28}}\).

**Average value: The average value of the function **\(f\)** is defined on the interval **\({\rm{(a, b)}}\)** is **\({f_{ave}} = \frac{1}{{b - a}}\int_a^b f (x)dx\).

The given function is \(g(x) = \sqrt(3){x}\) in the interval \({\rm{(1,8)}}\).

Obtain the average value of the function, \(g(x) = \sqrt(3){x}\) as follows.

The limit values are got from the interval \({\rm{(1,8)}}\) as \(a = 1\) and \(b = 8\).

Substitute \(a = 1\), \(b = 8\) and \(g(x)\) in the formula \({f_{ave}} = \frac{1}{{b - a}}\int_a^b f (x)dx\) and compute the average value of \(g(x)\) as follows.

\(\begin{aligned}{l}{g_{ave}} &= \frac{1}{{b - a}}\int_a^b g (x)dx\\ &= \frac{1}{{8 - 1}}\int_1^8 {\sqrt(3){x}} dx\\ &= \frac{1}{7}\int_1^8 {{x^{\frac{1}{3}}}} dx\end{aligned}\)

Further simplify as follows.

\(\begin{aligned}{c}{g_{ave}} &= \frac{1}{7}\int_1^8 {{x^{\frac{1}{3}}}} dx\\ &= \frac{1}{7}\left( {\frac{{{x^{\frac{4}{3}}}}}{{\left( {\frac{4}{3}} \right)}}} \right)_1^8\end{aligned}\)

\(\begin{aligned}{c}{g_{ave}} &= \frac{1}{7} \times \frac{3}{4}\left( {{8^{\frac{4}{3}}} - {1^{\frac{4}{3}}}} \right)\\ &= \frac{{45}}{{28}}\end{aligned}\)

Thus, the average value of the function \(g(x) = \sqrt(3){x}\) in the interval \({\rm{(1,8)}}\) is \(\frac{{45}}{{28}}\).

94% of StudySmarter users get better grades.

Sign up for free