Approximation of Definite Integral

Oct 13, 2025 by windailys.com 35 views

Introduction

The given formula, $\int_a^{a+h} f(x) \, dx \approx h f(a+h) - \frac{h^2}{2} f^{\prime}(a)$ , is an approximation of the definite integral of a function $f(x)$ over the interval $[a, a+h]$ . This formula is a part of the Taylor series expansion and is used to approximate the value of a definite integral. In this article, we will discuss the maximum degree of polynomials for which this formula is exact.

Taylor Series Expansion

The Taylor series expansion of a function $f(x)$ around a point $a$ is given by:

f(x) = f(a) + f^{\prime}(a)(x-a) + \frac{f^{\prime\prime}(a)}{2!}(x-a)^2 + \frac{f^{\prime\prime\prime}(a)}{3!}(x-a)^3 + \ldots

This expansion is used to approximate the value of a function at a point $x$ near $a$ . The Taylor series expansion can be used to approximate the value of a function at a point $x$ by truncating the series at a certain term.

Approximation of Definite Integral

The definite integral of a function $f(x)$ over the interval $[a, a+h]$ can be approximated using the Taylor series expansion. The approximation is given by:

\int_a^{a+h} f(x) \, dx \approx h f(a+h) - \frac{h^2}{2} f^{\prime}(a)

This approximation is obtained by integrating the Taylor series expansion of the function $f(x)$ over the interval $[a, a+h]$ .

Exactness of the Formula for Polynomials

The given formula is exact for polynomials of degree 1. This means that if the function $f(x)$ is a polynomial of degree 1, then the approximation of the definite integral is exact.

To prove this, let us consider a polynomial of degree 1, $f(x) = ax + b$ . The Taylor series expansion of this polynomial around the point $a$ is given by:

f(x) = f(a) + f^{\prime}(a)(x-a) = a(x-a) + b

The definite integral of this polynomial over the interval $[a, a+h]$ is given by:

\int_a^{a+h} f(x) \, dx = \int_a^{a+h} (a(x-a) + b) \, dx

Evaluating this integral, we get:

\int_a^{a+h} f(x) \, dx = \frac{ah^2}{2} + bh

Now, let us substitute the approximation formula into this expression:

h f(a+h) - \frac{h^2}{2} f^{\prime}(a) = h(a(a+h) + b) - \frac{h^2}{2} a

Simplifying this expression, we get:

h f(a+h) - \frac{h^2}{2} f^{\prime}(a) = \frac{ah^2}{2} + bh

This shows that the approximation formula is exact for polynomials of degree 1.

Higher Degree Polynomials

For polynomials of degree greater than 1, the approximation formula is not exact. To see this, let us consider a polynomial of degree 2, $f(x) = ax^2 + bx + c$ . The Taylor series expansion of this polynomial around the point $a$ is given by:

f(x) = f(a) + f^{\prime}(a)(x-a) + \frac{f^{\prime\prime}(a)}{2!}(x-a)^2 = a(x-a)^2 + b(x-a) + c

The definite integral of this polynomial over the interval $[a, a+h]$ is given by:

\int_a^{a+h} f(x) \, dx = \int_a^{a+h} (a(x-a)^2 + b(x-a) + c) \, dx

Evaluating this integral, we get:

\int_a^{a+h} f(x) \, dx = \frac{a h^3}{3} + \frac{b h^2}{2} + c h

Now, let us substitute the approximation formula into this expression:

h f(a+h) - \frac{h^2}{2} f^{\prime}(a) = h(a(a+h)^2 + b(a+h) + c) - \frac{h^2}{2} (2a)

Simplifying this expression, we get:

h f(a+h) - \frac{h^2}{2} f^{\prime}(a) = \frac{a h^3}{3} + \frac{b h^2}{2} + c h - a h^2

This shows that the approximation formula is not exact for polynomials of degree 2.

Conclusion

In conclusion, the given formula, $\int_a^{a+h} f(x) \, dx \approx h f(a+h) - \frac{h^2}{2} f^{\prime}(a)$ , is exact for polynomials of degree 1. This means that if the function $f(x)$ is a polynomial of degree 1, then the approximation of the definite integral is exact. However, for polynomials of degree greater than 1, the approximation formula is not exact.

References

[1] Taylor, B. (1715). Methodus Incrementorum Directa et Inversa.
[2] Euler, L. (1740). Institutiones Calculi Integralis.
[3] Cauchy, A. (1823). Cours d'Analyse de l'École Royale Polytechnique.

Q&A

Q: What is the maximum degree of polynomials for which the given formula is exact? A: The given formula is exact for polynomials of degree 1.

Q: Why is the formula not exact for polynomials of degree greater than 1? A: The formula is not exact for polynomials of degree greater than 1 because it does not take into account the higher-order terms of the Taylor series expansion.

Q: What is the Taylor series expansion of a function $f(x)$ around a point $a$ ? A: The Taylor series expansion of a function $f(x)$ around a point $a$ is given by:

f(x) = f(a) + f^{\prime}(a)(x-a) + \frac{f^{\prime\prime}(a)}{2!}(x-a)^2 + \frac{f^{\prime\prime\prime}(a)}{3!}(x-a)^3 + \ldots

Q: How is the definite integral of a function $f(x)$ over the interval $[a, a+h]$ approximated using the Taylor series expansion? A: The definite integral of a function $f(x)$ over the interval $[a, a+h]$ is approximated using the Taylor series expansion by integrating the series expansion over the interval $[a, a+h]$ .

Q: What is the approximation formula for the definite integral of a function $f(x)$ over the interval $[a, a+h]$ ? A: The approximation formula for the definite integral of a function $f(x)$ over the interval $[a, a+h]$ is given by:

\int_a^{a+h} f(x) \, dx \approx h f(a+h) - \frac{h^2}{2} f^{\prime}(a)

Q: Why is the approximation formula useful? A: The approximation formula is useful because it provides a simple and efficient way to approximate the value of a definite integral.

Q: What are some common applications of the approximation formula? A: The approximation formula has many common applications in mathematics, physics, and engineering, including:

Approximating the value of a definite integral
Solving differential equations
Modeling physical systems
Optimizing functions

Q: What are some limitations of the approximation formula? A: The approximation formula has several limitations, including:

It is only exact for polynomials of degree 1
It does not take into account higher-order terms of the Taylor series expansion
It may not be accurate for functions with sharp peaks or discontinuities

Conclusion

In conclusion, the given formula, $\int_a^{a+h} f(x) \, dx \approx h f(a+h) - \frac{h^2}{2} f^{\prime}(a)$ , is a useful approximation formula for the definite integral of a function $f(x)$ over the interval $[a, a+h]$ . However, it has several limitations, including being only exact for polynomials of degree 1 and not taking into account higher-order terms of the Taylor series expansion.

References

[1] Taylor, B. (1715). Methodus Incrementorum Directa et Inversa.
[2] Euler, L. (1740). Institutiones Calculi Integralis.
[3] Cauchy, A. (1823). Cours d'Analyse de l'École Royale Polytechnique.