Question

Let f(x) =
0 if x < 4
5 if -4 ≤ x < -1
-4 if -1 ≤ x < 5
0 if x ≥ 5

g(x) = ∫ [upper limit x, lower limit -4] f(t)dt

Determine g(-5), g(−3), g(0), and g(6).
Also, the absolute maximum of g(x) occurs when x = ? and and is the value ?

Answer 1

draw the graph of f(t). It is just a pair of horizontal lines,
one at f=5 and one at f = -4
The integral is just the algebraic sum of the areas of rectangles:
g(-5) = ∫[-4,-5] = 0
g(-3) = ∫[-4,-3] = 5*1 = 5
g(0) = ∫[-4,0] = ∫[-4,-1] + ∫[-1,0] = 5*3 - 4*1 = 11
g(6) = ∫[-4,6] = ∫[-4,-1] + ∫[-1,5] + ∫[5,6] = 5*3 - 4*6 + 0*1 = -9