We know that the function A(x) is defined as the integral:
[tex]A(x)=\int_2^xf(t)dt[/tex]
and that function f(t) is given in the graph.
a)
In this case we want to determine the value of A(3); according to the definition of the function A this will be given by:
[tex]A(3)=\int_2^3f(t)dt[/tex]
Now, geometrically this will mean that we would have to calculate the are under the curve of f(t) in the interval [2,3]; we show the area in the diagram below:
From it, we notice that this area is a triangle with base 1 and height 2; hence the area is:
[tex]A_1=\frac{1}{2}(2)(1)=1[/tex]
Therefore, we have that:
[tex]A(3)=1[/tex]
b)
Following the same procedure as before we have that:
[tex]A(7)=\int_2^7f(t)dt[/tex]
The area we need to calculate in this case is shown below:
From the previous step we have the first one; areas 2 and 3 area the areas of a triangle and rectangle, respectively.
Triangle 2 has a vase of 2 and a height of 3 while rectangle 3 has a length of 2 and a width of 3, then we have:
[tex]\begin{gathered} A_2=\frac{1}{2}(2)(3)=3 \\ A_3=(2)(3)=6 \end{gathered}[/tex]
Now, since areas 2 and 3 are below the x-axis this means that we that they are negative; with this in mind we have that the total area under the curve from 2 to 7 is:
[tex]A=1-3-6=-8[/tex]
And therefore:
[tex]A(7)=-8[/tex]
