Question

calculate the definite integral by referring to the figure with the indicated areas.
$$\int_{c}^{0} f(x) d x$$
area a = 1.571
area b = 2.826
area c = 5.221
area d = 1.871
$$\int_{c}^{0} f(x) d x=$$

Explanation:

Step1: Use the property of definite integral

We know that \(\int_{c}^{0}f(x)dx=\int_{c}^{d}f(x)dx+\int_{d}^{0}f(x)dx\).
Also, \(\int_{a}^{b}f(x)dx\) has the following sign rule: if the function \(y = f(x)\) is above the \(x -\)axis, \(\int_{a}^{b}f(x)dx>0\); if it is below the \(x -\)axis, \(\int_{a}^{b}f(x)dx<0\).
\(\int_{c}^{d}f(x)dx=- \text{Area of }D\) (since the region \(D\) is below the \(x -\)axis) and \(\int_{d}^{0}f(x)dx=\text{Area of }C\) (since the region \(C\) is above the \(x -\)axis).

Step2: Calculate the value of the integral

By the sum - rule of definite integrals \(\int_{c}^{0}f(x)dx=\int_{c}^{d}f(x)dx+\int_{d}^{0}f(x)dx\).
Substitute the values: \(\int_{c}^{d}f(x)dx=- 1.871\) (because area \(D = 1.871\) and the region \(D\) is below the \(x -\)axis) and \(\int_{d}^{0}f(x)dx = 5.221\) (because area \(C=5.221\) and the region \(C\) is above the \(x -\)axis).
Then \(\int_{c}^{0}f(x)dx=-1.871 + 5.221\).

Answer:

\(3.35\)