Question

find the exact value of the integral using formulas from geometry.
\\(\int_{3}^{5} (3 + 2x) dx\\)

\\(\int_{3}^{5} (3 + 2x) dx = \square\\) (simplify your answer.)

Explanation:

Step1: Identify the function type

The integrand $3 + 2x$ is a linear function ($y = mx + b$).

Step2: Find endpoints at x=3 and x=5

At $x=3$: $y = 3 + 2(3) = 9$
At $x=5$: $y = 3 + 2(5) = 13$

Step3: Calculate area of trapezoid

Area formula: $\frac{1}{2}(y_1 + y_2)(x_2 - x_1)$
Substitute values: $\frac{1}{2}(9 + 13)(5 - 3)$

Step4: Simplify the expression

$\frac{1}{2}(22)(2) = 22$

Answer:

22