What is the area of the region between the graphs of f(x)=x^(2)+12 x and g(x)=3x^(2)+10 from x=1 to x=4 ? Choose 1 answer: (A) (64)/(3) (B) 77 (C) 45 (D) 18

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What is the area of the region between the graphs of  f(x)=x^(2)+12 x and  g(x)=3x^(2)+10 from  x=1 to  x=4 ? Choose 1 answer: (A)  (64)/(3) (B) 77 (C) 45 (D) 18

Bytelearn · Accepted Answer

Find Difference of Functions: To find the area between two curves, we need to integrate the difference between the functions over the given interval. The difference between the functions f(x) and g(x) is: f(x) - g(x) = (x^2 + 12x) - (3x^2 + 10)Simplify the Expression: Simplify the expression to find the integrand: f(x) - g(x) = x^2 + 12x - 3x^2 - 10 f(x) - g(x) = -2x^2 + 12x - 10Integrate the Function: Now we integrate the function -2x^2 + 12x - 10 from x=1 to x=4: ∫ from 1 to 4 of (-2x^2 + 12x - 10) dxFind Antiderivative: Find the antiderivative of -2x^2 + 12x - 10: Antiderivative = (-2/3)x^3 + (12/2)x^2 - 10x Antiderivative = (-2/3)x^3 + 6x^2 - 10xEvaluate Antiderivative: Evaluate the antiderivative from x=1 to x=4: [(-2/3)(4)^3 + 6(4)^2 - 10(4)] - [(-2/3)(1)^3 + 6(1)^2 - 10(1)]Calculate Values: Calculate the values: [(-2/3)(64) + 6(16) - 40] - [(-2/3)(1) + 6(1) - 10] [(-128/3) + 96 - 40] - [(-2/3) + 6 - 10]Simplify the Expression: Simplify the expression: [-128/3 + 288/3 - 120/3] - [-2/3 + 18/3 - 30/3] [-128/3 + 288/3 - 120/3] - [-2/3 + 18/3 - 30/3] [-128 + 288 - 120] / 3 - [-2 + 18 - 30] / 3 [40/3] - [-14/3]Combine Terms: Combine the terms: 40/3 + 14/3 54/3Final Value: Simplify the final value: 54/3 = 18

What is the area of the region between the graphs of $f(x)=x^{2}+12 x$ and $g(x)=3 x^{2}+10$ from $x=1$ to $x=4$ ? $\newline$ Choose $1$ answer: $\newline$ (A) $\frac{64}{3}$ $\newline$ (B) $77$ $\newline$ (C) $45$ $\newline$ (D) $18$

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What is the area of the region between the graphs of f(x)=x2+12x f(x)=x^{2}+12 x f(x)=x2+12x and g(x)=3x2+10 g(x)=3 x^{2}+10 g(x)=3x2+10 from x=1 x=1 x=1 to x=4 x=4 x=4 ?\newlineChoose 111 answer:\newline(A) 643 \frac{64}{3} 364​\newline(B) 777777\newline(C) 454545\newline(D) 181818

What is the area of the region between the graphs of $f(x)=x^{2}+12 x$ and $g(x)=3 x^{2}+10$ from $x=1$ to $x=4$ ? $\newline$ Choose $1$ answer: $\newline$ (A) $\frac{64}{3}$ $\newline$ (B) $77$ $\newline$ (C) $45$ $\newline$ (D) $18$