Question

calculate the definite integral
a
int_{c}^{a} f(x) d x
by referring to the figure on
the right with the indicated areas.
int_{c}^{a} f(x) d x=quad
(simplify your answer.)

Explanation:

Step1: Use the property of definite integral

The definite integral \(\int_{c}^{a}f(x)dx=-\int_{a}^{c}f(x)dx\). And \(\int_{a}^{c}f(x)dx=\int_{a}^{b}f(x)dx+\int_{b}^{c}f(x)dx\).
The area above the \(x -\)axis is positive and the area below the \(x -\)axis is negative. So \(\int_{a}^{b}f(x)dx=-A\) (since the region \(A\) is below the \(x -\)axis), \(\int_{b}^{c}f(x)dx = B - C\) (where \(B\) is above the \(x -\)axis and \(C\) is below the \(x -\)axis).

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

\(\int_{a}^{c}f(x)dx=-A + B - C\).

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

Since \(\int_{c}^{a}f(x)dx=-\int_{a}^{c}f(x)dx\), then \(\int_{c}^{a}f(x)dx=A - B + C\).

Answer:

\(A - B + C\)