Ch. 5 - Integration

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 5 - Integration

Problem 5.2.51b

Chapter 5, Problem 5.2.51b

Properties of integrals Use only the fact that ∫₀⁴ 3𝓍 (4 ―𝓍) d𝓍 = 32, and the definitions and properties of integrals, to evaluate the following integrals, if possible.
(b) ∫₀⁴ 𝓍(𝓍 ― 4) d(𝓍)

Verified step by step guidance

Step 1: Begin by analyzing the given integral ∫₀⁴ 𝓍(𝓍 ― 4) d𝓍. Notice that the integrand 𝓍(𝓍 ― 4) can be rewritten as a product of terms. Expand the expression 𝓍(𝓍 ― 4) to simplify it into a polynomial form.

Step 2: Expand the integrand: 𝓍(𝓍 ― 4) = 𝓍² ― 4𝓍. This simplifies the integral to ∫₀⁴ (𝓍² ― 4𝓍) d𝓍.

Step 3: Use the linearity property of integrals to split the integral into two separate integrals: ∫₀⁴ (𝓍² ― 4𝓍) d𝓍 = ∫₀⁴ 𝓍² d𝓍 ― ∫₀⁴ 4𝓍 d𝓍.

Step 4: Factor out constants where applicable. For the second term, factor out the constant 4: ∫₀⁴ 𝓍² d𝓍 ― 4∫₀⁴ 𝓍 d𝓍.

Step 5: Evaluate each integral using the definitions and properties of integrals. Recall that ∫₀⁴ 3𝓍(4 ― 𝓍) d𝓍 = 32 is given, and use this information to relate the results if necessary. Apply the power rule for integration to compute ∫₀⁴ 𝓍² d𝓍 and ∫₀⁴ 𝓍 d𝓍.

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Key Concepts

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

Definite Integrals

A definite integral represents the signed area under a curve between two points on the x-axis. It is denoted as ∫ₐᵇ f(x) dx, where 'a' and 'b' are the limits of integration. The value of a definite integral can be interpreted as the accumulation of quantities, such as area, over the interval [a, b]. Understanding how to evaluate definite integrals is crucial for solving problems involving areas and accumulated quantities.

Properties of Integrals

The properties of integrals, such as linearity, additivity, and the ability to change variables, are essential for simplifying and evaluating integrals. For instance, the linearity property states that ∫(c * f(x)) dx = c * ∫f(x) dx for a constant 'c'. Additionally, the additivity property allows us to split integrals over adjacent intervals, which can be useful in evaluating more complex integrals by breaking them down into simpler parts.

Integration by Substitution

Integration by substitution is a technique used to simplify the process of evaluating integrals by changing the variable of integration. This method involves substituting a new variable for a function of the original variable, which can make the integral easier to solve. It is particularly useful when dealing with composite functions or when the integrand can be expressed in a simpler form through substitution.

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Related Practice

Textbook Question

{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 sin (πt² ) dt ( a Fresnel integral)

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Textbook Question

Using properties of integrals Use the value of the first integral I to evaluate the two given integrals.

I = ∫₀¹ (𝓍³ ― 2𝓍) d𝓍 = ―3/4

(b) ∫₁⁰ (2𝓍―𝓍³) d𝓍

Textbook Question

{Use of Tech} Approximating net area The following functions are positive and negative on the given interval.

f(x) = sin 2x on [0,3π/4]

(b) Approximate the net area bounded by the graph of f and the x-axis on the interval using a left, right, and midpoint Riemann sum with n = 4.

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Textbook Question

Properties of integrals Suppose ∫₀³ƒ(𝓍) d𝓍 = 2 , ∫₃⁶ƒ(𝓍) d𝓍 = ―5 , and ∫₃⁶g(𝓍) d𝓍 = 1. Evaluate the following integrals.

(b) ∫₃⁶ (―3g(𝓍)) d𝓍

Textbook Question

Working with area functions Consider the function ƒ and the points a, b, and c.

(b) Graph ƒ and A.

ƒ(𝓍) = 1/𝓍 ; a = 1 , b = 4 , c = 6

Textbook Question

The following functions are positive and negative on the given interval.

ƒ(𝓍) = xe⁻ˣ on [-1,1]

(b) Approximate the net area bounded by the graph of f and the x-axis on the interval using a left, right, and midpoint Riemann sum with n = 4.

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Properties of integrals Use only the fact that ∫₀⁴ 3𝓍 (4 ―𝓍) d𝓍 = 32, and the definitions and properties of integrals, to evaluate the following integrals, if possible.(b) ∫₀⁴ 𝓍(𝓍 ― 4) d(𝓍)

Verified video answer for a similar problem:

Key Concepts

Definite Integrals

Properties of Integrals

Integration by Substitution

Properties of integrals Use only the fact that ∫₀⁴ 3𝓍 (4 ―𝓍) d𝓍 = 32, and the definitions and properties of integrals, to evaluate the following integrals, if possible.
(b) ∫₀⁴ 𝓍(𝓍 ― 4) d(𝓍)