Domanda del libro di testo
In Exercises 75–78, find dy/dx.
y = ∫(from x to 1) (6/(3 + t^4))dt
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Ch. 5 - Integrals
Hass15th EditionThomas' CalculusISBN: 9780137616077
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TOC.tableOfContents
- 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 Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Capitolo 5, Problema 5.PE.9d
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
Guida verificata passo dopo passo1
Recall the property of definite integrals that allows constants to be factored out: for any constant \(c\), \(\int_a^b c \cdot h(x) \, dx = c \int_a^b h(x) \, dx\).
Apply this property to the integral \(\int_{-2}^5 (-\pi g(x)) \, dx\), rewriting it as \(-\pi \int_{-2}^5 g(x) \, dx\).
Use the given value \(\int_{-2}^5 g(x) \, dx = 2\) and substitute it into the expression to get \(-\pi \times 2\).
Express the integral in terms of \(\pi\) and the known value without calculating the numerical product, as per instructions.
Thus, the value of \(\int_{-2}^5 (-\pi g(x)) \, dx\) is \(-2\pi\) times the integral of \(g(x)\) over \([-2,5]\), which is \(-2\pi\).
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Properties of Definite Integrals
Definite integrals have linearity properties, meaning the integral of a sum is the sum of the integrals, and constants can be factored out. For example, ∫[a to b] c·f(x) dx = c·∫[a to b] f(x) dx. This allows simplification of integrals involving constants or sums of functions.
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Percorso guidato
05:43
Definition of the Definite Integral
Given Integral Values and Interval Consistency
When solving problems involving definite integrals, it is crucial to pay attention to the integration limits. The given values correspond to specific intervals, and the integral's value depends on these limits. Ensuring the limits match the problem's integral is essential for correct evaluation.
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11:11Improper Integrals: Infinite Intervals
Integral of a Scalar Multiple of a Function
If a function g(x) is multiplied by a scalar constant k, the integral over an interval is k times the integral of g(x) over that interval. Formally, ∫[a to b] k·g(x) dx = k·∫[a to b] g(x) dx. This property simplifies calculations involving constants like -π.
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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
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Domanda del libro di testo
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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√x + √y = 1, x = 0, y = 0
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Domanda del libro di testo
Definite Integrals
In Exercises 5–8, express each limit as a definite integral. Then evaluate the integral to find the value of the limit. In each case, P is a partition of the given interval, and the numbers cₖ are chosen from the subintervals of P.
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∥P∥→0 k = 1
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Domanda del libro di testo
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.
b. ∫₁² g(x) dx
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