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?

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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?

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

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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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?