find \\( \\int_{0}^{4}(4-x) d x \\) by using the formula for the area of a triangle. \n\\(…

Explanation:

Step1: Analyze the function and its graph

The function (y = 4 - x) is a straight - line. When (x = 0), (y=4); when (y = 0), (x = 4). The definite integral (\int_{0}^{4}(4 - x)dx) represents the area between the line (y = 4 - x), the (x) - axis ((y = 0)), (x = 0) and (x = 4). This region is a right - triangle.

Step2: Apply the formula for the area of a triangle

The formula for the area of a triangle is (A=\frac{1}{2}\times base\times height). For the triangle formed by the line (y = 4 - x), (x = 0), (x = 4) and (y = 0), the base (b = 4) (the difference between (x = 4) and (x = 0)) and the height (h=4) (the value of (y) when (x = 0)). [A=\frac{1}{2}\times4\times4]

Answer:

(8)