Question

calculate the definite integral by referring to the figure with the indicated areas.
area ( a = 1.453 ) area ( c = 5.369 )
area ( b = 2.265 ) area ( d = 1.894 )
int_{b}^{0} f(x) d x=square

Explanation:

Step1: Use the property of definite integral and area relationship

The definite integral \(\int_{b}^{0}f(x)dx=-\int_{0}^{b}f(x)dx\). And \(\int_{0}^{b}f(x)dx\) is related to the areas. For the integral from \(b\) to \(0\), we know that \(\int_{b}^{0}f(x)dx\) can be calculated by considering the sign of the area. The area above the \(x -\)axis is positive and the area below the \(x -\)axis is negative. Here, \(\int_{b}^{0}f(x)dx=-( - \text{Area of }B+\text{Area of }C)\) (because when we reverse the limits of integration \(\int_{a}^{b}f(x)dx=-\int_{b}^{a}f(x)dx\) and for the interval from \(b\) to \(0\), the part from \(b\) to \(0\) (assuming \(b<0\)) has area of \(B\) below the \(x -\)axis (negative) and area of \(C\) above the \(x -\)axis (positive)).

Step2: Substitute the values of areas

We are given \(\text{Area of }B = 2.265\) and \(\text{Area of }C=5.369\). Then \(\int_{b}^{0}f(x)dx=-( - 2.265 + 5.369)\)

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Answer:

\(-3.104\)