Question

calculate the definite integral
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

We know that \(\int_{c}^{a}f(x)dx=-\int_{a}^{c}f(x)dx\), 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 \(\int_{a}^{b}f(x)dx = 2310\) (positive as area \(B\) is above \(x -\)axis), \(\int_{b}^{c}f(x)dx=- 3180\) (negative as area \(C\) is below \(x -\)axis).

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

\(\int_{a}^{c}f(x)dx=\int_{a}^{b}f(x)dx+\int_{b}^{c}f(x)dx=2310-3180=- 870\)

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

Since \(\int_{c}^{a}f(x)dx=-\int_{a}^{c}f(x)dx\), then \(\int_{c}^{a}f(x)dx=-(-870) = 870\)

Answer:

\(870\)