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) = sin 𝓍 ; a = 0 , b = π/2 , c = π

Verified step by step guidance

1. Step 1: Understand the problem. You are tasked with finding the area function A(𝓍) = ∫ₐˣ ƒ(t) dt using the Fundamental Theorem of Calculus. The given function is ƒ(𝓍) = sin(𝓍), and the lower limit of integration is a = 0.
2. Step 2: Recall the Fundamental Theorem of Calculus. It states that if ƒ(t) is continuous on [a, x], then the derivative of the area function A(𝓍) with respect to 𝓍 is equal to ƒ(𝓍). In other words, A'(𝓍) = ƒ(𝓍).
3. Step 3: To find A(𝓍), integrate ƒ(t) = sin(t) with respect to t from the lower limit a = 0 to the upper limit x. The integral of sin(t) is -cos(t). Therefore, A(𝓍) = -cos(t) evaluated from t = 0 to t = x.
4. Step 4: Apply the limits of integration. Substitute the upper limit x and the lower limit 0 into the expression for A(𝓍). This gives A(𝓍) = [-cos(x)] - [-cos(0)]. Simplify the expression using the fact that cos(0) = 1.
5. Step 5: The area function A(𝓍) is now expressed in terms of x. You can use this function to evaluate the area for specific values of x, such as b = π/2 and c = π, by substituting these values into A(𝓍).

Key Concepts

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 allows us to evaluate definite integrals and find area functions effectively.

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 this context, it is used to find the area function A(x) from the function f(t).

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 represents the accumulated area under the curve of f(t) from a to x. In this problem, A(x) = ∫ₐˣ f(t) dt will help us determine the area under the curve of sin(x) from 0 to x, which is essential for understanding the behavior of the function over the specified interval.