Question

calculate the definite integral by referring to the figure with the indicated areas.
int_{a}^{c} f(x) d x
area ( a = 1.293 ) area ( c = 5.683 )
area ( b = 2.467 ) area ( d = 1.837 )
int_{a}^{c} f(x) d x=square

Explanation:

Step1: Recall the property of definite integral

The definite integral \(\int_{a}^{c}f(x)dx=\int_{a}^{b}f(x)dx+\int_{b}^{c}f(x)dx\). Also, if the function \(y = f(x)\) is above the \(x -\)axis, the integral \(\int_{m}^{n}f(x)dx\) (where \(m\lt n\)) is positive and equal to the area between the curve \(y = f(x)\), the \(x -\)axis, \(x = m\) and \(x = n\). If the function \(y = f(x)\) is below the \(x -\)axis, the integral \(\int_{m}^{n}f(x)dx\) (where \(m\lt n\)) is negative and equal to \(-\) (the area between the curve \(y = f(x)\), the \(x -\)axis, \(x = m\) and \(x = n\)).

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

For \(\int_{a}^{b}f(x)dx\), since the area \(A\) is above the \(x -\)axis, \(\int_{a}^{b}f(x)dx=1.293\). For \(\int_{b}^{c}f(x)dx\), the area \(B\) is below the \(x -\)axis (\(\int_{b}^{c}f(x)dx=- 2.467\)) and the area \(C\) is above the \(x -\)axis (\(\int_{b}^{c}f(x)dx\) (for the part of area \(C\)) \(=5.683\)).

Step3: Compute \(\int_{a}^{c}f(x)dx\)

\(\int_{a}^{c}f(x)dx=\int_{a}^{b}f(x)dx+\int_{b}^{c}f(x)dx\). Substitute the values: \(\int_{a}^{c}f(x)dx=(1.293)+(-2.467 + 5.683)\).

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Answer:

\(4.509\)