Question

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

Explanation:

Step1: Use the property of definite integral

We know that \(\int_{d}^{b}f(x)dx=-\int_{b}^{d}f(x)dx\). And \(\int_{b}^{d}f(x)dx=\int_{b}^{c}f(x)dx+\int_{c}^{d}f(x)dx\).
Also, \(\int_{a}^{b}f(x)dx=-A\) (since the area \(A\) is below the \(x -\)axis), \(\int_{b}^{c}f(x)dx = B\) (area \(B\) is above the \(x -\)axis), \(\int_{c}^{d}f(x)dx=-C + D\) (area \(C\) is below the \(x -\)axis and area \(D\) is above the \(x -\)axis).

Step2: Calculate \(\int_{b}^{d}f(x)dx\)

\(\int_{b}^{d}f(x)dx=B+(-C + D)=B - C+D\)

Step3: Calculate \(\int_{d}^{b}f(x)dx\)

Since \(\int_{d}^{b}f(x)dx=-\int_{b}^{d}f(x)dx\), then \(\int_{d}^{b}f(x)dx=-(B - C + D)=-B + C - D\)

Answer:

\(-B + C - D\)