Question

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

\(44\)