Calculus question

Question

The function f, is shown in the graph below. Let F′(x)=f(x), and assume that F(0)=0.

Sketch the graph of y=F(x). Find the values of F(b) for b=6,7,8

6=

7=

8=

Answer 1

The problem says that function F is the integral of f, that is, the area under the function f is determining the function F. In other words, if we take integral from the equality F' = f, using the first and second fundamental theorems of calculus, we get

F(b) = F(b) - F(0) = ∫₀^b f(x) dx.

Since ∫₀^b f(x) dx means the area under the curve of f from 0 to b, from the graph of f given above, we see that

F(6) = 9 - 1 = 8

F(7) = 9 - 3 = 6

F(8) = 9 - 4 = 5

Note that the area above the x-axis considered as positive and the are under x-axis as negative.