Ch. 5 - Integration

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 5 - Integration

Problem 5.5.5

Chapter 5, Problem 5.5.5

When using a change of variables u = g(𝓍) to evaluate the definite integral ∫ₐᵇ ƒ(g(𝓍)) g' (𝓍) d(𝓍), how are the limits of integration transformed?

Verified step by step guidance

Recognize that when performing a substitution in a definite integral, the variable of integration changes from \( x \) to \( u = g(x) \).

The original integral is \( \int_a^b f(g(x)) g'(x) \, dx \). After substitution, \( dx \) is replaced by \( \frac{du}{g'(x)} \), but since \( du = g'(x) dx \), the integral becomes \( \int_{u(a)}^{u(b)} f(u) \, du \).

To find the new limits of integration, evaluate the substitution function \( g(x) \) at the original limits: the lower limit \( a \) transforms to \( u(a) = g(a) \), and the upper limit \( b \) transforms to \( u(b) = g(b) \).

Thus, the definite integral with respect to \( x \) from \( a \) to \( b \) is equivalent to the integral with respect to \( u \) from \( g(a) \) to \( g(b) \).

This change of limits ensures the integral remains consistent under the substitution and allows you to evaluate the integral in terms of \( u \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Change of Variables (Substitution) in Integration

This technique simplifies integrals by substituting a new variable u = g(x), transforming the integral into terms of u. It helps to rewrite complex integrals into more manageable forms by changing the variable of integration.

Derivative of the Substitution Function

When substituting u = g(x), the differential dx is replaced by du = g'(x) dx. This derivative g'(x) adjusts the integrand to maintain equivalence between the original and transformed integrals.

Transformation of Limits of Integration

In definite integrals, the original limits a and b in terms of x must be converted to new limits in terms of u by evaluating u = g(a) and u = g(b). This ensures the integral's bounds correspond correctly to the substituted variable.

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Related Practice

Textbook Question

{Use of Tech} Sigma notation for Riemann sums Use sigma notation to write the following Riemann sums. Then evaluate each Riemann sum using Theorem 5.1 or a calculator.

The midpoint Riemann sum for f(x) = x³ on [3,11] with n = 32.

Textbook Question

Variations on the substitution method Evaluate the following integrals.

∫ 𝓍/(√𝓍―4) d𝓍

Textbook Question

Suppose an object moves along a line at 15 m/s, for 0 ≤ t < 2 and at 25 m/s, for 2 ≤ t ≤ 5, where t is measured in seconds. Sketch the graph of the velocity function and find the displacement of the object for 0 ≤ t ≤ 5.

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Textbook Question

Variations on the substitution method Evaluate the following integrals.

∫ 𝓍/(∛𝓍 + 4) d𝓍

Textbook Question

Evaluate ∫₃⁸ ƒ ′(t) dt , where ƒ ′ is continuous on [3, 8], ƒ(3) = 4, and ƒ(8) = 20 .

Textbook Question

Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus

∫₁⁹ 2/(√𝓍) d𝓍

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