How do we explicitly define g(x)*g*(*x*) in terms of f(x)*f*(*x*)?

To explicitly define \( g(x) \) in terms of \( f(x) \), we start by analyzing the information given in the problem.

### Restating the Problem

We know that the area \( A(PQ) \) of the region bounded by the arc \( PQ \) (part of the curve \( y = f(x) \)) and the segment \( PQ \) is given by:

\[
A(PQ) = \frac{1}{2} (f(a) + f(b))(b - a)
\]

We also know that the area under the curve from \( a \) to \( b \) is given by the integral:

\[
\text{Area under the curve} = \int_a^b f(x) \, dx
\]

### Deriving the Relationship

1. **Area of the Region under the Curve Minus the Segment:**

   The area under the curve minus the area under the straight segment (a straight line connecting points \( P \) and \( Q \)) should equal the area given in the problem:

   \[
   \int_a^b f(x) \, dx - \frac{1}{2} (f(a) + f(b))(b - a) = \frac{1}{2} (f(a) + f(b))(b - a)
   \]

   However, this simplifies to:

   \[
   \int_a^b f(x) \, dx = (f(a) + f(b))(b - a)
   \]

2. **Relationship to \( g(x) \):**

   This equation suggests that \( g(x) \) might be defined as an integral involving \( f(x) \). Since the left-hand side is an integral of \( f(x) \) from \( a \) to \( b \), let's hypothesize that \( g(x) \) is related to this integral. Specifically, assume:

   \[
   g(x) = \int_{x_0}^x f(t) \, dt
   \]

   Here, \( x_0 \) is some fixed point, and \( t \) is a dummy variable for integration.

3. **Differentiating \( g(x) \):**

   To find \( g'(x) \), differentiate both sides of the equation \( g(x) = \int_{x_0}^x f(t) \, dt \) with respect to \( x \):

   \[
   g'(x) = f(x)
   \]

### Conclusion

The explicit definition of \( g(x) \) in terms of \( f(x) \) is:

\[
g(x) = \int_{x_0}^x f(t) \, dt
\]

This implies that \( g'(x) = f(x) \). Thus, \( g(x) \) is the antiderivative of \( f(x) \), starting from some initial point \( x_0 \).

Would you like to explore more about how this relationship is derived or its applications? Here are some related questions:

1. Why does \( g(x) \) take the form of an integral of \( f(x) \)?
2. How does the Fundamental Theorem of Calculus apply in this context?
3. What is the significance of the constant of integration when defining \( g(x) \)?
4. Can we explore examples where such a relationship is used in practical scenarios?
5. How does this definition of \( g(x) \) help in solving other calculus problems?

**Tip:** When working with integrals and derivatives, remember that defining functions in terms of their integrals often leads to important relationships like \( g'(x) = f(x) \).

Discover how to explicitly define g(x) in terms of f(x) with a detailed calculus problem solution. This problem involves integrating f(x) to find g(x) and explores the Fundamental Theorem of Calculus. Suitable for undergraduate students studying calculus.

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