Question

20 multiple choice 1 point use geometry to evaluate the definite integral. \\(\int_{0}^{3} (3 - x) dx\\) \\(\bigcirc\\) 4.5 \\(\bigcirc\\) 9 \\(\bigcirc\\) 18 \\(\bigcirc\\) 2.25

Explanation:

Step1: Identify the graph shape

The function \( y = 3 - x \) is a straight line. For \( x = 0 \), \( y = 3 \); for \( x = 3 \), \( y = 0 \). So the region under the curve from \( x = 0 \) to \( x = 3 \) is a triangle.

Step2: Calculate triangle area

The formula for the area of a triangle is \( \frac{1}{2} \times base \times height \). Here, base \( = 3 \) (from \( x = 0 \) to \( x = 3 \)) and height \( = 3 \) (the \( y \)-value at \( x = 0 \)). So area \( = \frac{1}{2} \times 3 \times 3 = 4.5 \).

Answer:

4.5 (corresponding to the option with 4.5)