[Technology Exercise] When solving Exercises 33-40, you may need to use a calculator or a computer.
Use numerical integration to estimate the value of
π = 4 ∫ (from 0 to 1) [ 1 / (1 + x²) ] dx.

검증된 단계별 안내

Recognize that the integral given is \( \int_0^1 \frac{1}{1 + x^2} \, dx \), which is related to the arctangent function and is used to estimate \( \pi \) by the formula \( \pi = 4 \times \int_0^1 \frac{1}{1 + x^2} \, dx \).

Choose a numerical integration method to approximate the integral, such as the Trapezoidal Rule, Simpson's Rule, or a numerical integration function on a calculator or computer.

Divide the interval \([0,1]\) into \(n\) subintervals of equal width \( \Delta x = \frac{1 - 0}{n} = \frac{1}{n} \).

Calculate the function values \( f(x) = \frac{1}{1 + x^2} \) at the endpoints and at each subinterval point \( x_i = i \Delta x \) for \( i = 0, 1, 2, ..., n \).

Apply the chosen numerical integration formula to approximate the integral \( \int_0^1 \frac{1}{1 + x^2} \, dx \), then multiply the result by 4 to estimate \( \pi \).

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주요 개념

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

Definite Integral

A definite integral calculates the accumulated area under a curve between two limits. In this problem, the integral from 0 to 1 of 1/(1 + x²) represents the area under the curve of the function 1/(1 + x²), which is related to the arctangent function and helps approximate π.

Numerical Integration

Numerical integration involves approximating the value of an integral using computational methods when an exact solution is difficult or unnecessary. Techniques like the trapezoidal rule or Simpson's rule estimate the area under a curve by summing areas of simpler shapes, useful for evaluating integrals like the one given.

Relationship Between Integral and π

The integral of 1/(1 + x²) from 0 to 1 equals arctan(1), which is π/4. Multiplying this integral by 4 yields π. This connection allows π to be approximated by evaluating the integral numerically, linking calculus with fundamental constants.

[Technology Exercise] When solving Exercises 33-40, you may need to use a calculator or a computer.Use numerical integration to estimate the value ofπ = 4 ∫ (from 0 to 1) [ 1 / (1 + x²) ] dx.