Question

calculate the definite integral by referring to the figure with the indicated areas.

$$\int_{c}^{0} f(x) d x$$

area ( a = 1.289 )
area ( b = 2.466 )
area ( c = 5.385 )
area ( d = 1.717 )

$$\int_{c}^{0} f(x) d x=$$

Explanation:

Step1: Use the property of definite integral

The definite integral \(\int_{c}^{0}f(x)dx\) can be interpreted as the net - signed area between the curve \(y = f(x)\) and the \(x\) - axis from \(x = c\) to \(x = 0\). Areas above the \(x\) - axis contribute positively and areas below the \(x\) - axis contribute negatively.

Step2: Analyze the areas

From the figure, when calculating \(\int_{c}^{0}f(x)dx\), the area \(C\) is above the \(x\) - axis and area \(D\) is below the \(x\) - axis.
The formula for the definite integral in terms of area is \(\int_{c}^{0}f(x)dx=\text{Area of region above }x-\text{axis}-\text{Area of region below }x-\text{axis}\)
Since \(\text{Area of region above }x-\text{axis}=\text{Area }C = 5.385\) and \(\text{Area of region below }x-\text{axis}=\text{Area }D=1.717\)

Step3: Calculate the value

\(\int_{c}^{0}f(x)dx = 5.385-1.717\)

Answer:

\(3.668\)