Question

calculate the definite integral by referring to the figure with the indicated areas.
int_{a}^{d} f(x) d x
area ( a=12 ) area ( c=63.788 )
area ( b=21.6 ) area ( d=16.9 )
int_{a}^{d} f(x) d x=square
(type an integer or a decimal rounded to three decimal places as needed.)

Explanation:

Step1: Recall the property of definite integral and area

The definite integral \(\int_{a}^{d}f(x)dx=\int_{a}^{b}f(x)dx+\int_{b}^{0}f(x)dx+\int_{0}^{d}f(x)dx\). The area above the \(x -\)axis is positive and the area below the \(x -\)axis is negative.

Step2: Substitute the areas

We know that \(\int_{a}^{b}f(x)dx=\text{Area }A = 12\), \(\int_{b}^{0}f(x)dx=-\text{Area }B=- 21.6\), \(\int_{0}^{d}f(x)dx=\text{Area }C = 63.788\).

Step3: Calculate the sum

\(\int_{a}^{d}f(x)dx=12-21.6 + 63.788\).

$$
LATEXBLOCK0
$$

Answer:

\(54.188\)