교과서 질문
{Use of Tech} Areas of regions Find the area of the region 𝑅 bounded by the graph of ƒ and the 𝓍-axis on the given interval. Graph ƒ and show the region 𝑅.
ƒ(𝓍) = 2 ― |𝓍| on [ ― 2 , 4]
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Ch. 5 - Integration
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
5장, 문제 5.5.46
Definite integrals Use a change of variables or Table 5.6 to evaluate the following definite integrals.
∫₀¹ 2e²ˣ d𝓍
검증된 단계별 안내1
Step 1: Recognize that the integral ∫₀¹ 2e²ˣ d𝓍 involves an exponential function. To simplify, use a substitution method. Let u = 2x, which implies du = 2 dx.
Step 2: Rewrite the integral in terms of u. Substitute dx = du/2 and adjust the limits of integration. When x = 0, u = 0; when x = 1, u = 2.
Step 3: The integral becomes ∫₀² eᵘ du after substitution. Notice that the constant factor 2 from the original integral is canceled by the 1/2 factor from dx = du/2.
Step 4: Use the formula for the integral of eᵘ, which is ∫ eᵘ du = eᵘ + C. Apply this formula to evaluate the integral ∫₀² eᵘ du.
Step 5: Evaluate the definite integral by substituting the limits of integration. Compute eᵘ at u = 2 and subtract eᵘ at u = 0 to find the result.
비슷한 문제에 대한 검증된 영상 답변:
이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
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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 defined by a function over a specific interval [a, b]. It is denoted as ∫_a^b f(x) dx, where f(x) is the function being integrated. The result of a definite integral is a number that quantifies the total accumulation of the function's values between the limits a and b.
추천 영상:
05:43
Definition of the Definite Integral
Change of Variables
The change of variables technique, also known as substitution, is a method 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 Jacobian determinant is often used to adjust for the change in variable limits and the differential.
추천 영상:
06:35Changing Geometries
Exponential Functions
Exponential functions are mathematical functions of the form f(x) = a * e^(bx), where e is the base of natural logarithms, approximately equal to 2.71828. These functions are characterized by their rapid growth or decay and are commonly encountered in calculus, particularly in integration and differential equations. Understanding their properties is essential for evaluating integrals involving exponential terms.
추천 영상:
6:13Exponential Functions
관련 실천
교과서 질문
Variations on the substitution method Evaluate the following integrals.
∫ y²/(y + 1)⁴ dy
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교과서 질문
Limits of sums Use the definition of the definite integral to evaluate the following definite integrals. Use right Riemann sums and Theorem 5.1.
∫₃⁷ (4𝓍 + 6) d𝓍
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교과서 질문
Identifying definite integrals as limits of sums Consider the following limits of Riemann sums for a function ƒ on [a,b]. Identify ƒ and express the limit as a definite integral.
n
lim ∑ 𝓍*ₖ (ln 𝓍*ₖ) ∆𝓍ₖ on [1,2]
∆ → 0 k=1
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교과서 질문
Definite integrals Use a change of variables or Table 5.6 to evaluate the following definite integrals.
∫₁³ ( 2ˣ / 2ˣ + 4 ) d𝓍
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교과서 질문
Definite integrals Use a change of variables or Table 5.6 to evaluate the following definite integrals.
∫π/₄^π/² (cos 𝓍) / (sin² 𝓍) d𝓍
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