question the function f(x)=5 - |1 + x| is graphed below. use geometric formulas to evaluate the following…

Explanation:

Step1: Split the integral interval

The function (y = 5-\vert1 + x\vert) has a vertex at (x=- 1). Split the integral (\int_{-6}^{4}(5-\vert1 + x\vert)dx) into (\int_{-6}^{-1}(5-( - (1 + x)))dx+\int_{-1}^{4}(5-(1 + x))dx), which simplifies to (\int_{-6}^{-1}(6 + x)dx+\int_{-1}^{4}(4 - x)dx). Geometrically, the graph of (y = 5-\vert1 + x\vert) forms two right - angled triangles on the intervals ([-6,-1]) and ([-1,4]).

Step2: Calculate the area of the first triangle

For the interval ([-6,-1]), the base of the triangle (b_1=\vert-1-(-6)\vert = 5), and the height (h_1) of the triangle at (x=-1) for (y = 5-\vert1+( - 1)\vert=5) and at (x=-6) for (y = 5-\vert1+( - 6)\vert=0), so (h_1 = 5). The area of a triangle (A=\frac{1}{2}bh), so (A_1=\frac{1}{2}\times5\times5=\frac{25}{2}).

Step3: Calculate the area of the second triangle

For the interval ([-1,4]), the base of the triangle (b_2=\vert4-( - 1)\vert=5), and the height (h_2) of the triangle at (x=-1) for (y = 5) and at (x = 4) for (y=5-\vert1 + 4\vert=0), so (h_2 = 5). The area of the second triangle (A_2=\frac{1}{2}\times5\times5=\frac{25}{2}).

Step4: Find the value of the integral

The value of the definite integral (\int_{-6}^{4}(5-\vert1 + x\vert)dx) is the sum of the areas of the two triangles. So (A = A_1+A_2=\frac{25}{2}+\frac{25}{2}=25).

Answer:

25