Textbook Question
Area functions The graph of ƒ is shown in the figure. Let A(x) = ∫₀ˣ ƒ(t) dt and F(x) = ∫₂ˣ ƒ(t) dt be two area functions for ƒ. Evaluate the following area functions.
(d) F(8)
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9. Graphical Applications of Integrals
Area Between Curves
3:34 minutesProblem 5.3.18a
Textbook Question
Area functions for the same linear function Let ƒ(t) = 2t ― 2 and consider the two area functions A (𝓍) = ∫₁ˣ ƒ(t) dt and F(𝓍) = ∫₄ˣ ƒ(t) dt .
(a) Evaluate A (2) and A (3). Then use geometry to find an expression for A (𝓍) , for 𝓍 ≥ 1 .
Verified step by step guidance1
Step 1: Understand the problem. We are given a linear function ƒ(t) = 2t - 2 and two area functions A(𝓍) = ∫₁ˣ ƒ(t) dt and F(𝓍) = ∫₄ˣ ƒ(t) dt. The goal is to evaluate A(2) and A(3), and then find a general expression for A(𝓍) using geometry for 𝓍 ≥ 1.
Step 2: To evaluate A(2) and A(3), compute the definite integrals. For A(2), calculate ∫₁² (2t - 2) dt. For A(3), calculate ∫₁³ (2t - 2) dt. Use the formula for the definite integral of a linear function: ∫ (at + b) dt = (a/2)t² + bt + C.
Step 3: Apply the limits of integration to the results from Step 2. For example, after integrating ƒ(t), substitute the upper limit (𝓍 = 2 or 𝓍 = 3) and subtract the value of the integral at the lower limit (𝓍 = 1). This will give the values of A(2) and A(3).
Step 4: To find a general expression for A(𝓍) for 𝓍 ≥ 1, use geometry. The graph of ƒ(t) = 2t - 2 is a straight line. The area under the curve from t = 1 to t = 𝓍 forms a trapezoid or triangle, depending on the value of 𝓍. Use the formula for the area of a trapezoid or triangle to express A(𝓍) geometrically.
Step 5: Write the general expression for A(𝓍) based on the geometric interpretation. For example, if the area forms a trapezoid, use the formula A = (1/2)(base1 + base2)(height). If it forms a triangle, use A = (1/2)(base)(height). Ensure the expression is valid for 𝓍 ≥ 1.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Definite Integrals
Definite integrals represent the signed area under a curve defined by a function over a specific interval. In this context, A(x) and F(x) are defined as the definite integrals of the function ƒ(t) from different lower limits to x. Evaluating these integrals involves calculating the area under the curve from the lower limit to the upper limit, which is essential for finding A(2) and A(3).
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Area Under a Linear Function
The function ƒ(t) = 2t - 2 is a linear function, which means its graph is a straight line. The area under a linear function can be calculated using geometric shapes, such as triangles and rectangles. For the area function A(x), understanding the geometric interpretation allows for easier evaluation and expression of the area as a function of x, particularly for x ≥ 1.
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Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus links the concept of differentiation and integration, stating that if F is an antiderivative of f on an interval, then the definite integral of f from a to b can be computed as F(b) - F(a). This theorem is crucial for evaluating the area functions A(x) and F(x) since it allows us to find the values of these integrals by determining the antiderivative of the function ƒ(t) and applying the limits of integration.
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