The instructions for the integrals in Exercises 1–10 have three parts, one for the Midpoint Rule, one for the Trapezoidal Rule, and one for Simpson’s Rule.
II. Using the Trapezoidal Rule
b. Evaluate the integral directly and find |ET|.
∫ from 1 to 3 of (2x - 1) dx

검증된 단계별 안내

First, identify the function to integrate: \(f(x) = 2x - 1\), and the interval of integration: from \(a = 1\) to \(b = 3\).

To evaluate the integral directly, set up the definite integral: \(\int_{1}^{3} (2x - 1) \, dx\).

Find the antiderivative of the function \(f(x)\). Since \(f(x) = 2x - 1\), its antiderivative is \(F(x) = x^{2} - x\).

Evaluate the antiderivative at the bounds and subtract: calculate \(F(3) - F(1)\), which gives the exact value of the integral.

To find the trapezoidal rule error bound \(|E_{T}|\), use the formula \(|E_{T}| \leq \frac{(b - a)^{3}}{12n^{2}} \max_{a \leq x \leq b} |f''(x)|\), where \(n\) is the number of subintervals and \(f''(x)\) is the second derivative of \(f(x)\). Compute \(f''(x)\) and substitute the values to find \(|E_{T}|\).

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

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

Definite Integral

A definite integral calculates the exact area under a curve between two limits. It is represented as ∫ from a to b of f(x) dx, where a and b are the interval bounds. Evaluating it directly involves finding the antiderivative and computing its difference at the bounds.

Trapezoidal Rule

The Trapezoidal Rule is a numerical method to approximate definite integrals by dividing the area under the curve into trapezoids. It estimates the integral by summing the areas of these trapezoids, which are formed by connecting points on the function with straight lines.

Trapezoidal Rule Error Bound (|ET|)

The error bound |ET| measures the difference between the exact integral and the Trapezoidal Rule approximation. It depends on the second derivative of the function and the width of the subintervals, providing a way to estimate the accuracy of the numerical approximation.

The instructions for the integrals in Exercises 1–10 have three parts, one for the Midpoint Rule, one for the Trapezoidal Rule, and one for Simpson’s Rule.II. Using the Trapezoidal Ruleb. Evaluate the integral directly and find |ET|.∫ from 1 to 3 of (2x - 1) dx