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Ch. 5 - Integration
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Briggs 3rd EditionCh. 5 - Integration문제 5.5.54
5장, 문제 5.5.54

Definite integrals Use a change of variables or Table 5.6 to evaluate the following definite integrals.                                                                                                                         
                                                                                                                                                                              
 ∫π/₁₆^π/⁸ 8 csc² 4𝓍 d𝓍

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Step 1: Recognize that the integral involves the function csc²(4𝓍), which has a standard antiderivative. The antiderivative of csc²(u) is -cot(u).
Step 2: Perform a substitution to simplify the integral. Let u = 4𝓍, which implies that du = 4 d𝓍. Rewrite the integral in terms of u.
Step 3: Adjust the limits of integration according to the substitution. When 𝓍 = π/₁₆, u = 4(π/₁₆) = π/₄. When 𝓍 = π/₈, u = 4(π/₈) = π/₂.
Step 4: Rewrite the integral using the substitution. The integral becomes ∫π/₄^π/₂ 2 csc²(u) du, where the factor of 2 comes from dividing by 4 in the substitution.
Step 5: Evaluate the integral using the antiderivative of csc²(u). Substitute the limits of integration into -2 cot(u) and simplify.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Definite Integrals

A definite integral represents the signed area under a curve between two specified limits. It is denoted as ∫[a, b] f(x) dx, where 'a' and 'b' are the lower and upper limits, respectively. The result of a definite integral is a numerical value that quantifies the accumulation of the function's values over the interval [a, b].
추천 영상:
05:43
Definition of the Definite Integral

Change of Variables

Change of variables, or substitution, is a technique used in integration to simplify the integral by transforming it into a more manageable form. This involves substituting a new variable for an existing one, which can make the integral easier to evaluate. The process requires adjusting the limits of integration and the differential accordingly to maintain the integrity of the integral.
추천 영상:
06:35
Changing Geometries

Cosecant Function

The cosecant function, denoted as csc(x), is the reciprocal of the sine function, defined as csc(x) = 1/sin(x). In calculus, it often appears in integrals involving trigonometric functions. Understanding its properties and behavior is essential for evaluating integrals that include csc²(x), which is commonly encountered in integration problems.
추천 영상:
6:22
Graphs of Secant and Cosecant Functions
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교과서 질문

Approximating area from a graph Approximate the area of the region bounded by the graph (see figure) and the 𝓍-axis by dividing the interval [1, 7] into n = 6 subintervals. Use a left and right Riemann sum to obtain two different approximations.                                                                                                                                                                         

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교과서 질문

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


∫¹₁/₂ (t⁻³ ― 8) dt

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교과서 질문

Derivatives of integrals Simplify the following expressions.


d/dy ∫¹⁰ᵧ³ √(𝓍⁶ + 1) d𝓍

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교과서 질문

Average values Find the average value of the following functions on the given interval. Draw a graph of the function and indicate the average value.

ƒ(𝓍) = 1/(𝓍² + 1) on [―1, 1]

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교과서 질문

Areas of regions Find the area of the following regions.                                                                                                                   

                                                                                                                                                                 The region bounded by the graph of ƒ(𝓍) = (𝓍―4)⁴ and the 𝓍-axis between and 𝓍 = 2 and 𝓍= 6

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교과서 질문

Integrals with sin² 𝓍 and cos² 𝓍 Evaluate the following integrals.                                                                                                             

                                                                                                                                                                    

 ∫₀^π/⁴ cos² 8θ dθ

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