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Ch. 5 - Integration
Briggs - Calculus: Early Transcendentals 3rd Edition
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Tutti i libri di testoBriggs 3rd EditionCh. 5 - IntegrationProblema 5.3.15a
Capitolo 5, Problema 5.3.15a

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.
fig
ƒ(t) = 5 , a = 0

Guida verificata passo dopo passo
1
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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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.
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Definition of the Definite Integral

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.
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Finding Area When Bounds Are Not Given

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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Exponential Functions
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(a) Assuming ƒ is nonnegative on [a, b], draw a figure to illustrate the geometric meaning of these inequalities. Discuss your conclusions. b. 

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The velocity in ft/s of an object moving along a line is given by v = ƒ(t) on the interval 0 ≤ t ≤ 8 (see figure), where t is measured in seconds.

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Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

(a) If ƒ is a constant function on the interval [a,b], then the right and left Riemann sums give the exact value of ∫ₐᵇ ƒ(𝓍) d𝓍, for any positive integer n.

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