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

find \\( \\int _ { 0 } ^ { 4 } ( 4 - x ) d x \\) by using the formula for the area of a triangle. \n\\( \\int _ { 0 } ^ { 4 } ( 4 - x ) d x = \\) (simplify your answer.)

find \\( \\int _ { 0 } ^ { 4 } ( 4 - x ) d x \\) by using the formula for the area of a triangle. \n\\( \\int _ { 0 } ^ { 4 } ( 4 - x ) d x = \\) (simplify your answer.)

Answer

Explanation:

Step1: Find the line equation and intercepts

The function (y = 4 - x) is a straight - line. When (x = 0), (y=4) (y - intercept). When (y = 0), (0=4 - x), so (x = 4) (x - intercept).

Step2: Use the area formula of a triangle

The formula for the area of a triangle is (A=\frac{1}{2}\times base\times height). Here, the base is the length along the x - axis from (x = 0) to (x = 4), so (base=4). The height is the y - value when (x = 0), so (height = 4). Then (A=\frac{1}{2}\times4\times4).

Answer:

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