Working with area functions Consider the function ƒ and the points a, b, and c.
(a) Find the area function A (𝓍) = ∫ₐˣ ƒ(t) dt using the Fundamental Theorem.
ƒ(𝓍) = ― 12𝓍 (𝓍―1) (𝓍― 2) ; a = 0 , b = 1 , c = 2

검증된 단계별 안내

Step 1: Recall the Fundamental Theorem of Calculus, which states that if A(𝓍) = ∫ₐˣ ƒ(t) dt, then A'(𝓍) = ƒ(𝓍). This means the area function A(𝓍) is the antiderivative of ƒ(t) evaluated from a to 𝓍.

Step 2: Write down the given function ƒ(𝓍) = -12𝓍(𝓍 - 1)(𝓍 - 2). To find A(𝓍), we need to integrate ƒ(t) with respect to t over the interval [a, 𝓍], where a = 0.

Step 3: Substitute ƒ(t) = -12t(t - 1)(t - 2) into the integral A(𝓍) = ∫ₐˣ ƒ(t) dt. This becomes A(𝓍) = ∫₀ˣ -12t(t - 1)(t - 2) dt.

Step 4: Expand the polynomial -12t(t - 1)(t - 2) to simplify the integrand. Multiply the terms step by step to get a single polynomial expression in terms of t.

Step 5: Integrate the resulting polynomial term by term with respect to t. Use the power rule for integration, ∫tⁿ dt = (tⁿ⁺¹)/(n+1), and evaluate the definite integral from 0 to 𝓍 to find A(𝓍).

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주요 개념

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Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus links the concept of differentiation with integration. It states that if a function is continuous on an interval, then the integral of its derivative over that interval can be computed using the values of the function at the endpoints. This theorem is essential for finding area functions, as it allows us to express the area under a curve as an integral.

Definite Integral

A definite integral represents the signed area under a curve defined by a function over a specific interval [a, b]. It is calculated as the limit of Riemann sums and provides a numerical value that corresponds to the total accumulation of the function's values between the two bounds. In the context of area functions, it is used to compute the area from a starting point 'a' to a variable endpoint 'x'.

Area Function

An area function A(x) is defined as the integral of a function f(t) from a constant lower limit 'a' to a variable upper limit 'x'. It quantifies the area under the curve of f(t) from 'a' to 'x'. This function is crucial for understanding how the area changes as 'x' varies, and it is often used in applications involving accumulation and total quantities.

Working with area functions Consider the function ƒ and the points a, b, and c.(a) Find the area function A (𝓍) = ∫ₐˣ ƒ(t) dt using the Fundamental Theorem.ƒ(𝓍) = ― 12𝓍 (𝓍―1) (𝓍― 2) ; a = 0 , b = 1 , c = 2