To calculate the definite integral $\int_a^c f(x)dx$, we use the concept of signed areas as indicated by the diagram. The total area from $a$ to $c$ can be computed as follows:

1. Interpret the areas:

   • Area $A = 1.42$ (positive).
   • Area $B = 2.607$ (negative because it is below the x-axis).
   • Area $C = 5.291$ (positive).

2. Sum the signed areas: $\int_a^c f(x)dx = A - B + C$

3. Substitute values: $\int_a^c f(x)dx = 1.42 - 2.607 + 5.291$

4. Calculate: $\int_a^c f(x)dx = 4.104$

Thus, the value of the integral is: $\boxed{4.104}$

Would you like more details about the steps? Here are some related questions:

1. What is the role of signed areas in definite integrals?
2. How does the placement of an area below the x-axis affect integration?
3. Can this method apply to more complex curves?
4. How do we interpret these areas geometrically?
5. Could this process involve negative definite integrals?

Tip: Always note whether an area lies above or below the x-axis to determine its sign in integration.