Question

To find the blue shaded area above, we would calculate

Answer 1

According to the given graph, a and b (that are the limits of the integral) are 1 and 3 respectively, this is because the area under the curve is being evaluated from 1 to 3.

It means that a=1 and b=3.

f(x) is the function of which we are calculating the integral. That function is a linear function. We can find it by finding the equation of the line:

`m=(1-3)/(3-1)=(-2)/(2)=-1`
`y=-x+4`

It means that f(x)=-x+4.

The area can be found using the definite integral:

`\begin{gathered} \int_1^3(-x+4)dx \\ -(x^2)/(2)+4x \\ (-(3^2)/(2)+4(3))-(-(1^2)/(2)+4(1)) \\ (-(9)/(2)+12)-(-(1)/(2)+4) \\ -(9)/(2)+(1)/(2)+12-4 \\ -4+12-4 \\ 4 \end{gathered}`

It means that the area=4.