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Ch. 5 - Integration
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 5 - IntegrationProblema 5.3.101b
Capitolo 5, Problema 5.3.101b

{Use of Tech} Functions defined by integrals Consider the function g, which is given in terms of a definite integral with a variable upper limit.
b) Calculate g'(𝓍)


g(𝓍) = ∫₀ˣ sin² t dt

Guida verificata passo dopo passo
1
Step 1: Recognize that the function g(𝓍) is defined as a definite integral with a variable upper limit. This is a classic application of the Fundamental Theorem of Calculus, which states that if g(𝓍) = ∫ₐˣ f(t) dt, then g'(𝓍) = f(𝓍), provided f is continuous.
Step 2: Identify the integrand in the given function g(𝓍). Here, the integrand is sin²(t). This means that g(𝓍) = ∫₀ˣ sin²(t) dt.
Step 3: Apply the Fundamental Theorem of Calculus. According to the theorem, the derivative of g(𝓍) with respect to 𝓍 is simply the integrand evaluated at the upper limit of integration. Therefore, g'(𝓍) = sin²(𝓍).
Step 4: Confirm that the lower limit of integration (0 in this case) does not affect the derivative because it is a constant. The derivative depends only on the variable upper limit and the integrand.
Step 5: Write the final expression for g'(𝓍). Based on the steps above, g'(𝓍) = sin²(𝓍). This completes the differentiation process.

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Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus connects differentiation and integration, stating that if a function is defined as an integral with a variable upper limit, its derivative can be found by evaluating the integrand at that upper limit. Specifically, if g(x) = ∫ₐˣ f(t) dt, then g'(x) = f(x). This theorem is essential for calculating the derivative of functions defined by integrals.
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Fundamental Theorem of Calculus Part 1

Definite Integral

A definite integral represents the accumulation of quantities, such as area under a curve, between two specified limits. In the context of the function g(x) = ∫₀ˣ sin² t dt, the integral calculates the area under the curve of sin² t from 0 to x. Understanding how to evaluate definite integrals is crucial for applying the Fundamental Theorem of Calculus.
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Definition of the Definite Integral

Chain Rule

The Chain Rule is a fundamental principle in calculus used to differentiate composite functions. When applying the Chain Rule, if a function g(x) is defined in terms of another function, such as an integral, the derivative g'(x) can be found by multiplying the derivative of the outer function by the derivative of the inner function. This concept is particularly relevant when dealing with integrals that have variable limits.
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Intro to the Chain Rule
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{Use of Tech} Functions defined by integrals Consider the function g, which is given in terms of a definite integral with a variable upper limit.


(b) Calculate g'(𝓍)


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