Area functions for the same linear function Let ƒ(t) = t and consider the two area functions A(𝓍) = ∫₀ˣ ƒ(t) dt and F(𝓍) = ∫₂ˣ ƒ(t) dt .
(b) Evaluate F(4) and F(6). Then use geometry to find an expression for F (𝓍) , for 𝓍 ≥ 2. 

Verified step by step guidance

1. Step 1: Understand the problem. We are given a linear function ƒ(t) = t and two area functions: A(𝓍) = ∫₀ˣ ƒ(t) dt and F(𝓍) = ∫₂ˣ ƒ(t) dt. The goal is to evaluate F(4) and F(6), and then derive a general expression for F(𝓍) using geometry for 𝓍 ≥ 2.
2. Step 2: Recall the geometric interpretation of definite integrals. The integral ∫₂ˣ ƒ(t) dt represents the area under the curve ƒ(t) = t from t = 2 to t = 𝓍. Since ƒ(t) = t is a linear function, the graph is a straight line, and the area can be calculated as the area of a trapezoid or triangle.
3. Step 3: Evaluate F(4). To find F(4), compute the definite integral ∫₂⁴ t dt. This represents the area under the line ƒ(t) = t from t = 2 to t = 4. Use the formula for the definite integral of t, ∫ t dt = (1/2)t², and evaluate it at the bounds 2 and 4.
4. Step 4: Evaluate F(6). Similarly, compute the definite integral ∫₂⁶ t dt. This represents the area under the line ƒ(t) = t from t = 2 to t = 6. Again, use the formula for the definite integral of t, ∫ t dt = (1/2)t², and evaluate it at the bounds 2 and 6.
5. Step 5: Derive a general expression for F(𝓍) for 𝓍 ≥ 2. Using geometry, note that the area under the line ƒ(t) = t from t = 2 to t = 𝓍 forms a trapezoid or triangle. The formula for the area of a trapezoid is (1/2)(base1 + base2) × height. Alternatively, use the definite integral formula ∫₂ˣ t dt = (1/2)𝓍² - (1/2)(2²) to express F(𝓍).

Key Concepts

Definite Integral

A definite integral represents the signed area under a curve defined by a function over a specific interval. In this context, the area functions A(x) and F(x) are calculated using the definite integral of the function f(t) = t from a lower limit to an upper limit. Understanding how to evaluate definite integrals is crucial for finding the values of F(4) and F(6).

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus links the concept of differentiation with integration, stating that if F is an antiderivative of f on an interval, then the integral of f from a to b is equal to F(b) - F(a). This theorem is essential for evaluating the area functions A(x) and F(x) and helps in deriving expressions for these functions based on their limits.

Geometric Interpretation of Integrals

The geometric interpretation of integrals involves visualizing the area under a curve as a physical space that can be calculated. In this problem, using geometry to find an expression for F(x) for x ≥ 2 means recognizing the shape formed by the linear function f(t) = t and calculating the area of the resulting geometric figure, such as a triangle or trapezoid, to derive a formula for F(x).