Area functions Let A(x) be the area of the region bounded by the t -axis and the graph of y=ƒ(t) from t=0 to t=x. Consider the following functions and graphs.

b. Find A(6).

ƒ(t) =6 <IMAGE>

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Step 1: Understand the problem. We need to find the area A(x) under the curve y = f(t) from t = 0 to t = x, where f(t) = 6.

Step 2: Set up the integral. Since A(x) is the area under the curve from t = 0 to t = x, we can express it as an integral: A(x) = \( \int_{0}^{x} f(t) \, dt \).

Step 3: Substitute the function into the integral. Since f(t) = 6, the integral becomes: A(x) = \( \int_{0}^{x} 6 \, dt \).

Step 4: Evaluate the integral. The integral of a constant 6 with respect to t is 6t. So, A(x) = \( [6t]_{0}^{x} \).

Step 5: Apply the limits of integration. Substitute the limits into the evaluated integral: A(x) = 6x - 6(0). Therefore, A(x) = 6x. To find A(6), substitute x = 6 into the expression: A(6) = 6(6).

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주요 개념

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

Area Under a Curve

The area under a curve in calculus represents the integral of a function over a specified interval. For a function y = f(t), the area A(x) from t = 0 to t = x is calculated using the definite integral ∫ from 0 to x of f(t) dt. This concept is fundamental in understanding how to compute the total area bounded by the curve and the axes.

Definite Integral

A definite integral is a mathematical tool used to calculate the accumulation of quantities, such as area, over a specific interval. It is denoted as ∫ from a to b of f(t) dt, where a and b are the limits of integration. The result of a definite integral is a number that represents the net area between the function and the t-axis over the interval [a, b].

Function Evaluation

Function evaluation involves substituting a specific value into a function to determine its output. In the context of the problem, evaluating A(6) means calculating the area under the curve from t = 0 to t = 6 for the given function f(t). This process is essential for finding specific values related to the area function A(x).

Area functions Let A(x) be the area of the region bounded by the t -axis and the graph of y=ƒ(t) from t=0 to t=x. Consider the following functions and graphs.b. Find A(6).ƒ(t) =6 <IMAGE>