Question

calculate the definite integral by referring to the figure with the indicated areas.
\\\int_{a}^{c} f(x) d x\\
area \\( a=1.565 \\) area \\( c=5.533 \\) area \\( b=2.591 \\) area \\( d=1.65 \\)
\\\int_{a}^{c} f(x) d x=\square\\

Explanation:

Step1: Recall the property of definite integral and area

The definite integral \(\int_{a}^{c}f(x)dx=\int_{a}^{b}f(x)dx+\int_{b}^{c}f(x)dx\).
The area above the \(x -\)axis contributes positively and the area below the \(x -\)axis contributes negatively.
For \(\int_{a}^{b}f(x)dx\), since the area \(A\) is above the \(x -\)axis and area \(B\) is below the \(x -\)axis, \(\int_{a}^{b}f(x)dx=A - B\).
For \(\int_{b}^{c}f(x)dx\), since the area \(C\) is above the \(x -\)axis, \(\int_{b}^{c}f(x)dx = C\).

Step2: Calculate \(\int_{a}^{c}f(x)dx\)

We know that \(A = 1.565\), \(B=2.591\), \(C = 5.533\).
\(\int_{a}^{c}f(x)dx=(A - B)+C\).
Substitute the values: \((1.565-2.591)+5.533\).
First, calculate \(1.565-2.591=-1.026\).
Then, \(- 1.026+5.533 = 4.507\).

Answer:

\(4.507\)