Ch. 8 - Integration Techniques

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 8 - Integration Techniques

문제 8.8.24

8장, 문제 8.8.24

23-26. {Use of Tech} Simpson's Rule approximations. Find the indicated Simpson's Rule approximations to the following integrals.
24. ∫(4 to 8) √x dx using n = 4 and n = 8 subintervals

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Step 1: Understand Simpson's Rule. Simpson's Rule is a numerical method to approximate the value of a definite integral. It uses parabolic arcs to approximate the curve of the function. The formula is:

S = (h/3) [f(x₀) + 4f(x₁) + 2f(x₂) + 4f(x₃) + ... + f(xₙ)]

, where h is the width of each subinterval, and n is the number of subintervals (must be even).

Step 2: Determine the interval and subinterval width. The integral is over the interval [4, 8]. For n = 4 subintervals, calculate the width of each subinterval as

h = (8 - 4)/4 = 1

. For n = 8 subintervals, calculate

h = (8 - 4)/8 = 0.5

Step 3: Identify the x-values. For n = 4, the x-values are

x₀ = 4, x₁ = 5, x₂ = 6, x₃ = 7, x₄ = 8

. For n = 8, the x-values are

x₀ = 4, x₁ = 4.5, x₂ = 5, x₃ = 5.5, ..., x₈ = 8

Step 4: Evaluate the function at each x-value. The function is

f(x) = √x

. Compute

f(x₀), f(x₁), f(x₂), ...

for both n = 4 and n = 8 subintervals.

Step 5: Apply Simpson's Rule formula. Substitute the values of

h

f(x₀), f(x₁), f(x₂), ...

into the Simpson's Rule formula for both n = 4 and n = 8 subintervals. Simplify the expression to approximate the integral.

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주요 개념

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

Simpson's Rule

Simpson's Rule is a numerical method for approximating the definite integral of a function. It uses parabolic segments to estimate the area under a curve, providing a more accurate approximation than methods like the trapezoidal rule. The formula involves evaluating the function at equally spaced intervals and applying weights to these values, typically using three points for each segment.

Definite Integral

A definite integral represents the signed area under a curve defined by a function over a specific interval [a, b]. It is calculated using the Fundamental Theorem of Calculus, which connects differentiation and integration. The result of a definite integral is a number that quantifies the total accumulation of the function's values across the interval.

Subintervals

Subintervals are smaller segments into which the interval of integration is divided when applying numerical methods like Simpson's Rule. The number of subintervals, denoted as 'n', affects the accuracy of the approximation; more subintervals generally lead to a better approximation of the integral. Each subinterval's width is determined by dividing the total interval length by 'n'.

23-26. {Use of Tech} Simpson's Rule approximations. Find the indicated Simpson's Rule approximations to the following integrals.24. ∫(4 to 8) √x dx using n = 4 and n = 8 subintervals