Textbook Question
Derivatives of integrals Simplify the following expressions.
d/dy ∫¹⁰ᵧ³ √(𝓍⁶ + 1) d𝓍
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8. Definite Integrals
Fundamental Theorem of Calculus
1:41 minutesProblem 5.3.14g
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.
(g) F(2)
Verified step by step guidance1
Step 1: Understand the problem. We are tasked with evaluating F(2), where F(x) = ∫₂ˣ ƒ(t) dt. This represents the net area under the curve of ƒ(t) from t = 2 to t = x. Specifically, F(2) means evaluating the integral from t = 2 to t = 2.
Step 2: Recall a key property of definite integrals. When the upper and lower limits of integration are the same, the integral evaluates to 0. Mathematically, ∫ₐₐ ƒ(t) dt = 0 for any function ƒ(t).
Step 3: Apply this property to the given function. Since the limits of integration for F(2) are both 2, the integral simplifies to 0.
Step 4: Confirm this result conceptually. No area is enclosed when the start and end points of integration are identical, so the net area is 0.
Step 5: Conclude that F(2) = 0 based on the properties of definite integrals and the graph provided.
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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 calculated using the Fundamental Theorem of Calculus, which connects differentiation and integration. In this context, the area functions A(x) and F(x) are defined as integrals of the function f(t) over specified intervals, allowing us to compute the total area accumulated from the lower limit to x.
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Area Function
An area function, such as A(x) or F(x), quantifies the area under the curve of a function f(t) from a starting point to a variable endpoint x. For example, A(x) = ∫₀ˣ f(t) dt calculates the area from 0 to x, while F(x) = ∫₂ˣ f(t) dt starts from 2. These functions are useful for evaluating how the area changes as x varies, providing insights into the behavior of the function.
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Signed Area
Signed area refers to the concept that areas above the x-axis are considered positive, while areas below the x-axis are negative. This distinction is crucial when calculating definite integrals, as it affects the total area value. In the provided graph, the green area (8) is positive, while the red areas (5 and 11) are negative, which will influence the evaluation of the area functions F(2) and A(x) based on the intervals considered.
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