Area functions for linear 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 .

ƒ(t) = 2t + 5 , a = 0

Verified step by step guidance

1. Step 1: Understand the problem. The goal is to find the area function A(x) = ∫ₐˣ ƒ(t) dt, where ƒ(t) = 2t + 5 and a = 0. This represents the area under the curve of ƒ(t) from t = a to t = x.
2. Step 2: Set up the integral. Substitute the given function ƒ(t) = 2t + 5 into the integral: A(x) = ∫₀ˣ (2t + 5) dt.
3. Step 3: Break the integral into parts. Use the linearity of integration to separate the terms: A(x) = ∫₀ˣ 2t dt + ∫₀ˣ 5 dt.
4. Step 4: Compute each integral. For ∫₀ˣ 2t dt, use the power rule for integration: ∫ t^n dt = (t^(n+1))/(n+1). For ∫₀ˣ 5 dt, treat 5 as a constant and integrate: ∫ c dt = c * t.
5. Step 5: Combine the results and simplify. After evaluating the definite integrals, combine the terms to express A(x) as a function of x. This will give the area function A(x).

Key Concepts

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 area function A(x), which accumulates the area under the function f(t) from 'a' to 'x'.

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 total area under the curve of f(t) from 'a' to 'x'. In this case, with f(t) = 2t + 5, A(x) will yield a linear function representing the area as 'x' changes.

Graphing Area Functions

Graphing the area function A(x) involves plotting the accumulated area under the curve of f(t) as 'x' varies. The resulting graph typically shows how the area increases with 'x', reflecting the behavior of the original function. For linear functions like f(t) = 2t + 5, the area function will also be linear, making it easier to visualize the relationship between the two.