Area functions The graph of ƒ is shown in the figure. Let A(x) = ∫₀ˣ ƒ(t) dt and F(x) = ∫₂ˣ ƒ(t) dt be two area functions for ƒ. Evaluate the following area functions.
(a) A(2)

검증된 단계별 안내

Step 1: Understand the problem. We are tasked with evaluating A(2), where A(x) = ∫₀ˣ ƒ(t) dt. This represents the area under the curve of ƒ(t) from t = 0 to t = x.

Step 2: Analyze the graph. The graph shows the function ƒ(t) and the areas of specific regions. From t = 0 to t = 1, the area is shaded green and labeled as 'Area = 8'. This area is above the t-axis, so it contributes positively to the integral.

Step 3: From t = 1 to t = 2, the graph dips below the t-axis, and the area is shaded green and labeled as 'Area = 5'. Since this region is below the t-axis, it contributes negatively to the integral.

Step 4: To compute A(2), sum the contributions of the areas from t = 0 to t = 2. Specifically, add the positive area from t = 0 to t = 1 and subtract the negative area from t = 1 to t = 2.

Step 5: Write the expression for A(2): A(2) = (Area from t = 0 to t = 1) - (Area from t = 1 to t = 2). Substitute the values from the graph: A(2) = 8 - 5.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Definite Integral

A definite integral represents the signed area under a curve between two points on the x-axis. It is calculated using the integral symbol and limits of integration, providing a numerical value that corresponds to the net area, accounting for areas above and below the x-axis.

Area Function

An area function, such as A(x) = ∫₀ˣ ƒ(t) dt, defines the accumulated area under the curve of a function ƒ from a starting point to a variable endpoint x. This function helps in understanding how the area changes as x varies, and it is crucial for evaluating specific area values at given points.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus links differentiation and integration, stating that if F is an antiderivative of f on an interval, then the definite integral of f from a to b can be computed as F(b) - F(a). This theorem is essential for evaluating area functions and understanding their properties.

Area functions The graph of ƒ is shown in the figure. Let A(x) = ∫₀ˣ ƒ(t) dt and F(x) = ∫₂ˣ ƒ(t) dt be two area functions for ƒ. Evaluate the following area functions.(a) A(2)