Textbook Question
In Exercises 75–78, find dy/dx.
y = ∫(from x to 1) (6/(3 + t^4))dt
Ch. 5 - Integrals
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 5, Problem 5.PE.10d
If ∫₀² ƒ(x) dx = π, ∫₀² 7g(x) dx = 7, and ∫₀¹ g(x) dx = 2, find the value of each of the following.
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d. ∫₀² √2ƒ(x) dx
Verified step by step guidance1
Identify the given integral and the constant multiplier: the integral to find is \(\int_0^2 \sqrt{2} f(x) \, dx\).
Recall the property of integrals that allows constants to be factored out: \(\int_a^b c \cdot h(x) \, dx = c \int_a^b h(x) \, dx\) where \(c\) is a constant.
Apply this property to the integral: \(\int_0^2 \sqrt{2} f(x) \, dx = \sqrt{2} \int_0^2 f(x) \, dx\).
Use the given value \(\int_0^2 f(x) \, dx = \pi\) and substitute it into the expression: \(\sqrt{2} \times \pi\).
Express the final answer as \(\sqrt{2} \pi\), which represents the value of the integral \(\int_0^2 \sqrt{2} f(x) \, dx\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Definite Integral and Linearity
The definite integral represents the net area under a curve between two points. It is linear, meaning constants can be factored out: ∫a^b c·f(x) dx = c·∫a^b f(x) dx. This property allows simplification when integrating scaled functions.
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Definition of the Definite Integral
Given Integral Values and Substitution
Known integral values can be directly used to find new integrals involving the same function. For example, if ∫₀² f(x) dx = π, then ∫₀² √2 f(x) dx = √2 · π by applying linearity, avoiding the need for re-integration.
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Understanding Integral Limits
The limits of integration define the interval over which the function is integrated. Changing limits affects the integral's value, so it is important to note the interval when applying given integral values or combining integrals.
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Related Practice
Textbook Question
Find the areas of the regions enclosed by the curves and lines in Exercises 15–26.
y = sin x, y = x, 0 ≤ x ≤ π/4
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Textbook Question
Evaluate the integrals in Exercises 47–68.
∫₁² 4 dv
v²
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Textbook Question
Express the solutions of the initial value problems in Exercises 35 and 36 in terms of integrals.
dy/dx = sin x/x , y(5) = -3
Textbook Question
If ∫²₋₂ 3ƒ(x) dx = 12, ∫⁵₋₂ ƒ(x) dx = 6, and ∫⁵₋₂ g(x) dx = 2, find the value of each of the following.
d. ∫⁵₋₂ (-πg(x)) dx
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Textbook Question
Find the areas of the regions enclosed by the curves and lines in Exercises 15–26.
√x + √y = 1, x = 0, y = 0
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