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) = cos 𝓍 ; a = 0 , b = π/2 , c = π

Verified step by step guidance

1. 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 derivative of the area function A(𝓍) is equal to the original function ƒ(𝓍).
2. Step 2: To find the area function A(𝓍), integrate ƒ(t) = cos(t) with respect to t from the lower limit a = 0 to the upper limit 𝓍. The integral of cos(t) is sin(t).
3. Step 3: Apply the definite integral formula: A(𝓍) = ∫ₐˣ ƒ(t) dt = [sin(t)]ₐˣ. Substitute the limits of integration into the antiderivative.
4. Step 4: Substitute the lower limit a = 0 and the upper limit 𝓍 into the expression: A(𝓍) = sin(𝓍) - sin(0). Simplify the result using the fact that sin(0) = 0.
5. Step 5: The area function A(𝓍) is now expressed as A(𝓍) = sin(𝓍). This function represents the accumulated area under ƒ(t) = cos(t) from t = 0 to t = 𝓍.

Key Concepts

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus connects differentiation and integration, stating that if a function is continuous on an interval, then the integral of its derivative over that interval gives the net change of the function. Specifically, it allows us to evaluate definite integrals using antiderivatives, which is essential for finding area functions.

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 bounds, which is crucial for determining the area function A(x) in the given problem.

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. This function represents the accumulated area under the curve of f(t) from a to x, and it is essential for understanding how the area changes as x varies, particularly in the context of the problem involving the cosine function.