The accompanying figure shows the area of regions bounded by the graph of f and the​ x-axis. Evaluate the following integral.
Integral from 0 to a f left parenthesis x right parenthesis dx
12
4
10
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Part 1
Integral from 0 to a f left parenthesis x right parenthesis dxequals

​(Simplify your​ answer.)

To evaluate the definite integral \(\int_0^a f(x) \, dx\), we need to understand the areas given in the accompanying figure and how they relate to the integral.

The problem statement indicates that the integral's value can be determined using the areas provided. Typically, the figure would show areas of regions between the graph of \( f(x) \) and the \( x \)-axis.

Based on the areas provided:
- The area from \( x = 0 \) to some point along the \( x \)-axis is 12.
- The area from this point to another point along the \( x \)-axis is 4.
- The area from this second point to \( x = a \) is 10.

To solve the integral, we need to sum the areas bounded by the graph of \( f(x) \) and the \( x \)-axis from \( x = 0 \) to \( x = a \). 

Assuming the areas are positive and directly under the curve (above the \( x \)-axis), we add these areas:

\[
\int_0^a f(x) \, dx = 12 + 4 + 10
\]

Thus, 

\[
\int_0^a f(x) \, dx = 26
\]

So, the value of the integral \(\int_0^a f(x) \, dx\) is 26.

Would you like details on how definite integrals relate to area under a curve? If you have any other questions, feel free to ask. Here are some questions you might be interested in:

1. How to interpret definite integrals in terms of area?
2. How to find the area under a curve if part of the region is below the x-axis?
3. How to calculate definite integrals using the Fundamental Theorem of Calculus?
4. How to handle integrals with discontinuous functions?
5. What are the applications of definite integrals in real-life scenarios?

**Tip:** When interpreting areas under a curve, remember that areas above the x-axis contribute positively to the integral, while areas below the x-axis contribute negatively.

Learn how to evaluate the definite integral \(\int_0^a f(x) \, dx\) step-by-step. Understand the concept of area under a curve and use the Fundamental Theorem of Calculus to find the integral's value. Suitable for advanced high school students and those studying calculus.

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