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Ch. 5 - Integration
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 5 - IntegrationProblem 5.3.35
Chapter 5, Problem 5.3.35

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


βˆ«β‚βΉ 2/(βˆšπ“) d𝓍

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1
Step 1: Recognize that the integral βˆ«β‚βΉ 2/(βˆšπ“) d𝓍 can be rewritten using exponents. Recall that βˆšπ“ is equivalent to 𝓍^(1/2), so the integrand becomes 2 * 𝓍^(-1/2).
Step 2: Apply the power rule for integration. The general formula for integrating 𝓍^n is βˆ«π“^n d𝓍 = (𝓍^(n+1))/(n+1) + C, where n β‰  -1. Here, n = -1/2.
Step 3: Compute the antiderivative of 2 * 𝓍^(-1/2). Using the power rule, the antiderivative becomes 2 * (𝓍^(1/2))/(1/2), which simplifies to 4 * 𝓍^(1/2).
Step 4: Use the Fundamental Theorem of Calculus to evaluate the definite integral. Substitute the limits of integration (𝓍 = 1 and 𝓍 = 9) into the antiderivative. This gives [4 * βˆšπ“] evaluated from 1 to 9.
Step 5: Calculate the difference between the values of the antiderivative at the upper and lower limits. Specifically, compute 4 * √9 - 4 * √1. Simplify the expression to find the final result.

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Key Concepts

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

Definite Integrals

Definite integrals represent the signed area under a curve between two specified limits. They are calculated using the integral symbol with lower and upper bounds, indicating the interval over which the function is evaluated. The result of a definite integral is a numerical value that quantifies this area, which can be interpreted in various contexts, such as physics or economics.
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Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus links the concept of differentiation with integration, providing a method to evaluate definite integrals. It states that if a function is continuous on an interval, then the integral of its derivative over that interval equals the difference in the values of the original function at the endpoints. This theorem allows for the computation of definite integrals by finding an antiderivative of the integrand.
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Antiderivatives

An antiderivative of a function is another function whose derivative yields the original function. In the context of definite integrals, finding an antiderivative is crucial because it allows us to apply the Fundamental Theorem of Calculus. For example, if we need to evaluate the integral of a function, we first determine its antiderivative and then compute the difference between its values at the upper and lower limits of integration.
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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) = xΒ³ on [3,11] with n = 32.

Textbook Question

Definite integrals Use geometry (not Riemann sums) to evaluate the following definite integrals. Sketch a graph of the integrand, show the region in question, and interpret your result.                                                                                                                                      

                                                                                                                                                                                       

 βˆ«β‚€β΄ √(16― 𝓍² ) d𝓍

Textbook Question

Variations on the substitution method Evaluate the following integrals.                                                                                                        

                                                                                                                                                                    

 βˆ« π“/(βˆšπ“β€•4) d𝓍

Textbook Question

Variations on the substitution method Evaluate the following integrals.                                                                                                        

                                                                                                                                                                    

 βˆ« 𝓍/(βˆ›π“ + 4) d𝓍

Textbook Question

Evaluate βˆ«β‚ƒβΈ Ζ’ β€²(t) dt , where Ζ’ β€² is continuous on [3, 8], Ζ’(3) = 4, and Ζ’(8) = 20 .

Textbook Question

When using a change of variables u = g(𝓍) to evaluate the definite integral βˆ«β‚α΅‡ Ζ’(g(𝓍)) g' (𝓍) d(𝓍), how are the limits of integration transformed?