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The image you've uploaded contains a mathematical problem involving integrals and areas. The problem describes a function \( y = F(x) \) which is the antiderivative of \( y = f(x) \). The graph shows two regions labeled \( A \) and \( B \) under the curve \( y = f(x) \), and the areas of these regions are given by:

\[
S_A = 5, \quad S_B = 2.
\]

Additionally, it mentions that \( F(-2) = 1 \), and the task is to find \( F(1) \).

To solve this, we can use the relationship between a function and its antiderivative. Since \( F(x) \) is the antiderivative of \( f(x) \), we know:

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

Let’s break down the solution step by step:

1. The area of region \( A \) corresponds to the integral from \( x = -2 \) to \( x = 0 \), so:

\[
S_A = \int_{-2}^0 f(x) \, dx = 5
\]

2. The area of region \( B \) corresponds to the integral from \( x = 0 \) to \( x = 1 \), so:

\[
S_B = \int_0^1 f(x) \, dx = 2
\]

3. From the properties of the antiderivative \( F(x) \), we can compute \( F(1) \) as:

\[
F(1) - F(-2) = \int_{-2}^1 f(x) \, dx
\]

This integral is the sum of the areas of regions \( A \) and \( B \):

\[
F(1) - 1 = S_A + S_B = 5 + 2 = 7
\]

So:

\[
F(1) = 1 + 7 = 8
\]

Thus, \( F(1) = 8 \).

Let me know if you'd like more details or further steps! 

Here are some follow-up questions to deepen the understanding:
1. How can the properties of definite integrals help in solving such problems?
2. What is the difference between a function and its antiderivative in terms of graphical interpretation?
3. How would the solution change if the area of region A were different, say \( S_A = 7 \)?
4. How does the Fundamental Theorem of Calculus apply here?
5. Can the value of \( F(x) \) be determined at other points given the same information?

Tip: When calculating areas using definite integrals, always break the region into smaller parts for clarity and easier computation.

Biết rằng y = F(x) là một nguyên hàm của hàm số y = f(x). Đồ thị của hàm số y = f(x) được biểu diễn trong hình bên dưới. Biết rằng diện tích các phần hình phẳng A và B lần lượt là SA = 5, SB = 2. Khi F(-2) = 1 hãy tính giá trị F(1).

This problem involves using antiderivatives and definite integrals to compute F(1), given that the areas under a curve between certain points are provided. The key concept used is the Fundamental Theorem of Calculus. Suitable for high school students studying calculus.

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