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Let
f be the function defined by
f(x)=sin((pi)/(4)x). What is the average value of
f on the interval
[(4)/(3),(8)/(3)] written in simplest form?

Let $f$ be the function defined by $f(x)=\sin \left(\frac{\pi}{4} x\right)$ . What is the average value of $f$ on the interval $\left[\frac{4}{3}, \frac{8}{3}\right]$ written in simplest form?

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Write variable expressions for arithmetic sequences

Full solution

Q. Let

f

be the function defined by

f(x)=\sin \left(\frac{\pi}{4} x\right)

. What is the average value of

f

on the interval

\left[\frac{4}{3}, \frac{8}{3}\right]

written in simplest form?

Calculate difference: To find the average value of a continuous function $f(x)$ on the interval $[a, b]$ , we use the formula: $\newline$ Average value = $\frac{1}{(b-a)} \int_{a}^{b} f(x) \, dx$ $\newline$ Here, $f(x) = \sin\left(\frac{\pi}{4}x\right)$ , $a = \frac{4}{3}$ , and $b = \frac{8}{3}$ .
Set up integral: First, we calculate the difference $b - a$ : $b - a = \frac{8}{3} - \frac{4}{3} = \frac{4}{3}$
Simplify coefficient: Now, we set up the integral to find the average value: $\newline$ Average value = $\frac{1}{\left(\frac{4}{3}\right)}$ * $\int_{\frac{4}{3}}^{\frac{8}{3}} \sin\left(\frac{\pi}{4}x\right) dx$
Use substitution method: We can simplify the coefficient in front of the integral: $\newline$ $(1/((4)/(3))) = (3)/(4)$ $\newline$ So, Average value = $(3)/(4) \times \int_{(4)/(3)}^{(8)/(3)} \sin((\pi)/(4)x) \, dx$
Change limits of integration: To integrate $\sin\left(\frac{\pi}{4}x\right)$ , we use the substitution method. Let $u = \frac{\pi}{4}x$ , then $du = \frac{\pi}{4}dx$ , or $dx = \frac{4}{\pi}du$ .
Write integral in terms of $u$ : We also need to change the limits of integration. When $x = \frac{4}{3}$ , $u = \frac{\pi}{4}\cdot\frac{4}{3} = \frac{\pi}{3}$ . When $x = \frac{8}{3}$ , $u = \frac{\pi}{4}\cdot\frac{8}{3} = \frac{2\pi}{3}$ .
Simplify integral further: Now we can write the integral in terms of $u$ :
Average value = $\frac{3}{4} \times \int_{\frac{\pi}{3}}^{\frac{2\pi}{3}} \sin(u) \times \frac{4}{\pi} \, du$
Integrate $\sin(u)$ : We can simplify the integral further by taking out the constant $\frac{4}{\pi}$ :
Average value = $\frac{3}{4} \times \frac{4}{\pi} \times \int_{\frac{\pi}{3}}^{\frac{2\pi}{3}} \sin(u) \, du$
Evaluate antiderivative: Now we integrate $\sin(u)$ from $\frac{\pi}{3}$ to $\frac{2\pi}{3}$ : $\int \sin(u) \, du = -\cos(u) + C$ So, we need to evaluate $-\cos(u)$ from $\frac{\pi}{3}$ to $\frac{2\pi}{3}$ .
Evaluate bounds: Evaluating the antiderivative at the bounds gives us: $-\cos\left(\frac{2\pi}{3}\right) - \left(-\cos\left(\frac{\pi}{3}\right)\right)$
Multiply by coefficient: We know that $\cos\left(\frac{\pi}{3}\right) = \frac{1}{2}$ and $\cos\left(\frac{2\pi}{3}\right) = -\frac{1}{2}$ . So, $-(-\frac{1}{2}) - (-\frac{1}{2}) = \frac{1}{2} + \frac{1}{2} = 1$
Final average value: Now we multiply this result by the coefficient we found earlier: $\newline$ Average value = $(\frac{3}{4}) \times (\frac{4}{\pi}) \times 1$
Final average value: Now we multiply this result by the coefficient we found earlier: $\newline$ Average value = $(\frac{3}{4}) \times (\frac{4}{\pi}) \times 1$ Simplifying this expression gives us the final average value: $\newline$ Average value = $(\frac{3}{\pi})$

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