find the exact value of the integral using formulas from geometry. \n$$ int _ { 3 } ^ { 7 } ( 1 + 2 x ) d x…

find the exact value of the integral using formulas from geometry. \n$$ int _ { 3 } ^ { 7 } ( 1 + 2 x ) d x $$\n$$ int _ { 3 } ^ { 7 } ( 1 + 2 x ) d x = square $$ (simplify your answer.)

find the exact value of the integral using formulas from geometry. \n$$ int _ { 3 } ^ { 7 } ( 1 + 2 x ) d x $$\n$$ int _ { 3 } ^ { 7 } ( 1 + 2 x ) d x = square $$ (simplify your answer.)

Answer

Explanation:

Step1: Find the function values at endpoints

Let (y = 1+2x). When (x = 3), (y_1=1 + 2\times3=7). When (x = 7), (y_2=1+2\times7 = 15).

Step2: Calculate the integral using the trapezoid area formula

The integral (\int_{a}^{b}f(x)dx) (where (a = 3), (b = 7), (f(x)=1 + 2x)) is equivalent to the area of a trapezoid. The area formula of a trapezoid is (A=\frac{(y_1 + y_2)(b - a)}{2}). Substitute (y_1 = 7), (y_2=15), (a = 3), (b = 7) into the formula: (A=\frac{(7 + 15)(7 - 3)}{2}). First, calculate (7+15=22) and (7 - 3=4). Then (A=\frac{22\times4}{2}). Since (\frac{22\times4}{2}=22\times2).

Answer:

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