The average yalue of csc^(2)x over the interval from x=(pi)/(6) to x=(pi)/(4) is (A) (3sqrt3)/(pi) (B) (sqrt3)/(pi) (C) (12 )/(pi)(sqrt3-1) (D) 3(sqrt3-1)

Question

The average yalue of  csc^(2)x over the interval from  x=(pi)/(6) to  x=(pi)/(4) is (A)  (3sqrt3)/(pi) (B)  (sqrt3)/(pi) (C)  (12 )/(pi)(sqrt3-1) (D)  3(sqrt3-1)

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Understand the problem: Understand the problem. We need to find the average value of the function csc^(2)x over the interval [π/6, π/4]. The average value of a function f(x) over the interval [a, b] is given by the integral of f(x) from a to b, divided by the length of the interval (b - a).Write down the formula: Write down the formula for the average value of a function. The average value of a function f(x) over the interval [a, b] is given by: Average value = (1/(b - a)) * ∫[a to b] f(x) dxApply the formula to csc^(2)x: Apply the formula to csc^(2)x. For our function csc^(2)x, a = π/6 and b = π/4. So the average value is: Average value = (1/(π/4 - π/6)) * ∫[π/6 to π/4] csc^(2)x dxCalculate the length of the interval: Calculate the length of the interval. The length of the interval is π/4 - π/6 = (π/4)(3/3) - (π/6)(2/2) = (3π/12) - (2π/12) = π/12.Set up the integral: Set up the integral. Now we have: Average value = (1/(π/12)) * ∫[π/6 to π/4] csc^(2)x dx Average value = (12/π) * ∫[π/6 to π/4] csc^(2)x dxEvaluate the integral: Evaluate the integral. The integral of csc^(2)x is -cot(x). So we need to evaluate -cot(x) from π/6 to π/4.Calculate the antiderivative at the bounds: Calculate the antiderivative at the bounds. -cot(π/4) = -1 (since cot(π/4) = 1) -cot(π/6) = -√3 (since cot(π/6) = √3)Find the difference of the antiderivative at the bounds: Find the difference of the antiderivative at the bounds. (-1) - (-√3) = -1 + √3Multiply by the factor from step 5: Multiply by the factor from step 5. Average value = (12/π) * (-1 + √3) Average value = (12/π) * √3 - (12/π)Simplify the expression: Simplify the expression. The average value cannot be simplified further in terms of exact values. So, the average value of csc^(2)x over the interval [π/6, π/4] is (12/π) * √3 - (12/π).

The average value of $\csc^2 x$ over the interval from $x=\frac{\pi}{6}$ to $x=\frac{\pi}{4}$ is $\newline$ (A) $\frac{3\sqrt{3}}{\pi}$ $\newline$ (B) $\frac{\sqrt{3}}{\pi}$ $\newline$ (C) $\frac{12}{\pi}(\sqrt{3}-1)$ $\newline$ (D) $3(\sqrt{3}-1)$

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