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 .                                                                                                                               
                                                                                                                                                                               
 <IMAGE>                                                                                                                                                                                                           
                                                                                                                                                                                     
 ƒ(t) = 4t + 2 , a = 0

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, where ƒ(t) = 4t + 2 and a = 0. This involves integrating the given linear function ƒ(t) with respect to t, starting from the lower limit a = 0 to the upper limit x.
2. Step 2: Set up the integral. Write the area function as A(𝓍) = ∫₀ˣ (4t + 2) dt. This represents the accumulation of the area under the curve ƒ(t) = 4t + 2 from t = 0 to t = x.
3. Step 3: Perform the integration. Break the integral into two parts: ∫₀ˣ 4t dt and ∫₀ˣ 2 dt. Use the power rule for integration on the first term and the constant rule for integration on the second term. For the first term, ∫ 4t dt = 4 * (t² / 2) = 2t². For the second term, ∫ 2 dt = 2t.
4. Step 4: Apply the limits of integration. Substitute the limits of integration (from t = 0 to t = x) into the result of the integration. For the first term, evaluate 2t² at t = x and t = 0. For the second term, evaluate 2t at t = x and t = 0.
5. Step 5: Combine the results. After applying the limits, combine the results from both terms to express the area function A(𝓍). The final expression will be a function of x that represents the accumulated area under the curve ƒ(t) = 4t + 2 from t = 0 to t = x.

Key Concepts

Area Function

An area function, denoted as A(x), represents the accumulated area under a curve from a fixed point 'a' to a variable point 'x'. It is defined mathematically as A(x) = ∫ₐˣ f(t) dt, where f(t) is the function being integrated. This concept is crucial for understanding how the area changes as 'x' varies, and it provides insights into the behavior of the function over an interval.

Definite Integral

The definite integral is a fundamental concept in calculus that calculates the net area under a curve between two points on the x-axis. It is represented as ∫ₐˣ f(t) dt, where 'a' is the lower limit and 'x' is the upper limit. This integral not only quantifies the area but also accounts for the sign of the function, allowing for the determination of positive and negative areas.

Graphing Functions

Graphing functions involves plotting points on a coordinate system to visually represent the relationship between variables. For area functions, the graph illustrates how the area accumulates as 'x' increases. Understanding how to graph functions is essential for interpreting the behavior of the area function and analyzing its properties, such as increasing or decreasing trends.