Area functions for constant functions Consider the following functions ƒ and real numbers a (see figure).
(a) Find and graph the area function A(𝓍) = ∫ₐˣ ƒ(t) dt for ƒ.

ƒ(t) = 5 , a = 0

Verified step by step guidance

Step 1: Understand the problem. The function ƒ(t) = 5 is a constant function, and we are tasked with finding the area function A(𝓍) = ∫ₐˣ ƒ(t) dt, where a = 0. The graph shows a rectangle with height ƒ(t) = 5 and width determined by the interval [a, 𝓍].

Step 2: Recall the formula for the definite integral of a constant function. For a constant function ƒ(t) = c, the integral ∫ₐˣ ƒ(t) dt simplifies to c * (𝓍 - a). In this case, c = 5 and a = 0.

Step 3: Substitute the values into the formula. Replace c with 5 and a with 0 in the formula for the definite integral. This gives A(𝓍) = 5 * (𝓍 - 0).

Step 4: Simplify the expression for A(𝓍). The area function becomes A(𝓍) = 5𝓍, which represents the area of the rectangle as a function of 𝓍.

Step 5: Graph the area function A(𝓍). The graph of A(𝓍) = 5𝓍 is a straight line passing through the origin with a slope of 5. This represents how the area under the curve ƒ(t) = 5 grows linearly as 𝓍 increases.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Definite Integral

A definite integral represents the signed area under a curve between two points on the x-axis. It is denoted as ∫ₐˣ f(t) dt, where 'a' is the lower limit and 'x' is the upper limit. This concept is fundamental in calculating the total accumulation of quantities, such as area, over an interval.

Area Function

The area function A(x) is defined as the integral of a function f(t) from a fixed point 'a' to a variable point 'x'. It quantifies the area under the curve of f(t) from 'a' to 'x', providing a way to visualize how the area changes as 'x' varies. In this case, with f(t) = 5, A(x) will yield a linear function.

Constant Function

A constant function is a function that always returns the same value regardless of the input. In this scenario, f(t) = 5 is a constant function, meaning the height of the rectangle representing the area under the curve remains constant. This simplifies the calculation of the area, as it can be computed as the product of the base and height.

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