Find the Root Mean Square of the function f(x) = 2x+3 on the interval [1, 4].

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Accepted Answer

Step:1 ：Step 1: The formula for the Root Mean Square (RMS) of a function on the interval [a, b] is ( RMS = sqrt{frac{1}{b-a} int_{a}^{b} [f(x)]^2 dx} ). Here, f(x) = 2x + 3, a = 1, and b = 4.Step:2 ：Step 2: First, calculate ([f(x)]^2). This is ((2x + 3)^2 = 4x^2 + 12x + 9).Step:3 ：Step 3: Next, calculate the integral of ([f(x)]^2) from 1 to 4. This is ( int_{1}^{4} (4x^2 + 12x + 9) dx = [frac{4}{3}x^3 + 6x^2 + 9x]_{1}^{4} = frac{4}{3}*64 + 6*16 + 9*4 - (frac{4}{3}*1 + 6*1 + 9*1) = 256/3 + 96 + 36 - 20/3 = 348.)Step:4 ：Step 4: Finally, substitute these values into the RMS formula to get ( RMS = sqrt{frac{1}{4-1} * 348} = sqrt{116} approx 10.77.)

Answer

Step: 1 ：Step 1: The formula for the Root Mean Square (RMS) of a function on the interval [a, b] is ( RMS = sqrt{frac{1}{b-a} int_{a}^{b} [f(x)]^2 dx} ). Here, f(x) = 2x + 3, a = 1, and b = 4.Step: 2 ：Step 2: First, calculate ([f(x)]^2). This is ((2x + 3)^2 = 4x^2 + 12x + 9).Step: 3 ：Step 3: Next, calculate the integral of ([f(x)]^2) from 1 to 4. This is ( int_{1}^{4} (4x^2 + 12x + 9) dx = [frac{4}{3}x^3 + 6x^2 + 9x]_{1}^{4} = frac{4}{3}*64 + 6*16 + 9*4 - (frac{4}{3}*1 + 6*1 + 9*1) = 256/3 + 96 + 36 - 20/3 = 348.)Step: 4 ：Step 4: Finally, substitute these values into the RMS formula to get ( RMS = sqrt{frac{1}{4-1} * 348} = sqrt{116} approx 10.77.)

Find the Root Mean Square of the function f(x) = 2x+3 on the interval [1, 4].

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