Ch. 5 - Integration

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo

Briggs 3rd Edition

Ch. 5 - Integration

Problema 5.R.102e

Capitolo 5, Problema 5.R.102e

Function defined by an integral Let H (𝓍) = ∫₀ˣ √(4 ― t²) dt, for ― 2 ≤ 𝓍 ≤ 2.
(e) Find the value of s such that H (𝓍) = sH(―𝓍)

Guida verificata passo dopo passo

Step 1: Understand the problem. The function H(𝓍) is defined as an integral from 0 to 𝓍 of √(4 - t²) dt. The goal is to find the value of s such that H(𝓍) = sH(-𝓍). This involves symmetry properties of the integral and the function.

Step 2: Analyze the integrand √(4 - t²). This function is even because √(4 - t²) = √(4 - (-t)²). This means the integrand is symmetric about the y-axis.

Step 3: Use the property of definite integrals for even functions. For an even function f(t), ∫₀ˣ f(t) dt is equal to ∫₋ˣ⁰ f(t) dt. This symmetry will help relate H(𝓍) and H(-𝓍).

Step 4: Express H(-𝓍) using the definition of the integral. H(-𝓍) = ∫₀⁻ˣ √(4 - t²) dt. By reversing the limits of integration, this becomes H(-𝓍) = -∫₋ˣ⁰ √(4 - t²) dt.

Step 5: Combine the symmetry property and the reversed integral to find the relationship between H(𝓍) and H(-𝓍). Use this relationship to determine the value of s such that H(𝓍) = sH(-𝓍).

Risposta video verificata per un problema simile:

Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.

Durata del video:

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

Definite Integral

A definite integral represents the signed area under a curve defined by a function over a specific interval. In this case, H(𝓍) is defined as the integral of √(4 - t²) from 0 to 𝓍, which geometrically corresponds to the area under the curve of the function from the lower limit to the upper limit.

Video consigliato:

Percorso guidato

05:43

Definition of the Definite Integral

Symmetry in Functions

Symmetry in functions refers to the property where a function exhibits the same behavior when its input is negated. For the function H(𝓍), understanding its symmetry can help in finding the relationship between H(𝓍) and H(−𝓍), which is crucial for solving the equation H(𝓍) = sH(−𝓍).

Video consigliato:

Percorso guidato

06:21

Properties of Functions

Parameterization

Parameterization involves expressing a function in terms of a variable that can take on a range of values. In this context, the variable s acts as a parameter that scales the relationship between H(𝓍) and H(−𝓍). Finding the appropriate value of s requires analyzing how these two integrals relate to each other.

Video consigliato:

04:26

Parameterizing Equations Example 4

Pratica correlata

Domanda del libro di testo

Integration by Riemann sums Consider the integral ∫₁⁴ (3𝓍― 2) d𝓍.

(b) Use summation notation to express the right Riemann sum in terms of a positive integer n .

100

views

Domanda del libro di testo

Evaluating integrals Evaluate the following integrals.

∫₀¹ 𝓍 • 2ˣ²⁺¹ d𝓍

views

Domanda del libro di testo

Evaluating integrals Evaluate the following integrals.

∫₀¹ √𝓍 (√𝓍 + 1) d𝓍

views

Domanda del libro di testo

Limit definition of the definite integral Use the limit definition of the definite integral with right Riemann sums and a regular partition to evaluate the following definite integrals. Use the Fundamental Theorem of Calculus to check your answer.

∫₀² (𝓍²―4) d𝓍

155

views

Domanda del libro di testo

Use geometry and properties of integrals to evaluate the following definite integrals.

∫₀⁴ √(8𝓍―𝓍²) d𝓍 . (Hint: Complete the square .)

views

Domanda del libro di testo

Evaluating integrals Evaluate the following integrals.

∫ (cos 7ω) /(16 + sin² 7ω) dω

views