Find the area under the graph of f over the interval [-2,4]. f(x)={[x^(2)+6,x 2]:}

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Find the area under the graph of  f over the interval  [-2,4].  f(x)={[x^(2)+6,x <= 2],[5x,x > 2]:}

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Identify Intervals and Functions: Identify the intervals and corresponding functions for the piecewise function f(x). The function f(x) is defined as x^2 + 6 for x ≤ 2, and as 5x for x > 2. We need to find the area under the graph of f(x) from x = -2 to x = 4. This means we will have to split the integral at x = 2, where the definition of the function changes.Set Up Integral for First Interval: Set up the integral for the first interval from x = -2 to x = 2. For x ≤ 2, f(x) = x^2 + 6. We will integrate this function from -2 to 2. The integral is ∫ from -2 to 2 of (x^2 + 6) dx.Calculate Integral for First Interval: Calculate the integral for the first interval. ∫ from -2 to 2 of (x^2 + 6) dx = [x^3/3 + 6x] from -2 to 2. Evaluating this from -2 to 2 gives: (2^3/3 + 6*2) - ((-2)^3/3 + 6*(-2)).Perform Calculations for First Interval: Perform the calculations for the first interval. (2^3/3 + 6*2) - ((-2)^3/3 + 6*(-2)) = (8/3 + 12) - (-8/3 - 12). Simplify the expression: (8/3 + 12 + 8/3 + 12) = (16/3 + 24) = (16/3 + 72/3) = 88/3.Set Up Integral for Second Interval: Set up the integral for the second interval from x = 2 to x = 4. For x > 2, f(x) = 5x. We will integrate this function from 2 to 4. The integral is ∫ from 2 to 4 of 5x dx.Calculate Integral for Second Interval: Calculate the integral for the second interval. ∫ from 2 to 4 of 5x dx = [5x^2/2] from 2 to 4. Evaluating this from 2 to 4 gives: (5*4^2/2) - (5*2^2/2).Perform Calculations for Second Interval: Perform the calculations for the second interval. (5*4^2/2) - (5*2^2/2) = (5*16/2) - (5*4/2) = (40) - (10) = 30.Add Areas for Total Area: Add the areas from both intervals to find the total area under the graph of f(x) from x = -2 to x = 4. Total area = Area from first interval + Area from second interval. Total area = 88/3 + 30.Convert Second Area and Add: Convert the second area to a fraction with a common denominator and add the two areas. Total area = 88/3 + (30 * 3/3) = 88/3 + 90/3 = 178/3.Convert Total Area: Convert the total area to a mixed number or decimal if necessary. Total area = 178/3 = 59 1/3.

Find the area under the graph of $f$ over the interval $[-2,4]$ . $\newline$ $f(x)=\left\{\begin{array}{ll} x^{2}+6 & x \leq 2 \\ 5 x & x>2 \end{array}\right.$

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Find the area under the graph of $f$ over the interval $[-2,4]$ . $\newline$ $f(x)=\left\{\begin{array}{ll} x^{2}+6 & x \leq 2 \\ 5 x & x>2 \end{array}\right.$