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Calculus II · Unit 4B · lesson

Using Series to Approximate Definite Integrals

Concept

Learning objectives

integrate a power series to approximate a definite integral lacking an elementary antiderivative.

Using Series to Approximate Definite Integrals

Explanation

Power series can replace an unavailable antiderivative

Some important integrals cannot be expressed with elementary functions. A power series for the integrand can still be integrated term by term, producing a rapidly computable numerical approximation. The method is especially effective on a small interval where powers shrink quickly.

The approximation must include an error argument. This may come from alternating-series error, a remainder bound for the integrand series, or a bound on the integrated tail. The method is not numerical guesswork; it is exact symbolic transformation followed by controlled truncation.

Bridge

Integrate the polynomial approximation when no elementary antiderivative exists

Some important functions, such as e−x2e^{-x^2}, have no elementary antiderivative. A power series turns the integrand into an infinite polynomial, which can be integrated term by term on a suitable interval.

The method has three distinct stages: expand the integrand, integrate the series, and control the truncation error. The result is not merely a numerical trick; it creates a convergent representation of a genuinely new function.

Polynomial approximations make a nonelementary integral computable. The integrand and successive polynomial approximations over the integration interval.
Read this graph as text

Polynomial approximations make a nonelementary integral computable. The curve e to the negative x squared is compared with two truncated series on the interval from zero to one half, with the area under each shown. The integrand and successive polynomial approximations over the integration interval.

Written labels, distinct line styles, markers, and fill patterns communicate every relationship in polynomial approximations make a nonelementary integral computable; color is never the only cue.

Why it matters: The integrand and successive polynomial approximations over the integration interval.

Polynomial approximations make a nonelementary integral computable

The curve e to the negative x squared is compared with two truncated series on the interval from zero to one half, with the area under each shown.

Polynomial approximations make a nonelementary integral computable. The integrand and successive polynomial approximations over the integration interval.

Concept

Integrate the series, not the impossible antiderivative

If f(x)=∑cnxnf(x)=\sum c_nx^n on the interval of integration, then

∫0bf(x) dx=∑n=0∞cnbn+1n+1,\int_0^b f(x)\,dx=\sum_{n=0}^{\infty}\frac{c_n b^{n+1}}{n+1},

provided termwise integration is justified.

Guided walkthrough

Approximate a Gaussian-type integral

Since

e−x2=1−x2+x42!−x63!+⋯ ,e^{-x^2}=1-x^2+\frac{x^4}{2!}-\frac{x^6}{3!}+\cdots,

we obtain

∫00.5e−x2 dx=0.5−0.533+0.5510−0.5742+⋯ .\int_0^{0.5}e^{-x^2}\,dx =0.5-\frac{0.5^3}{3}+\frac{0.5^5}{10}-\frac{0.5^7}{42}+\cdots.

The series alternates with rapidly shrinking terms.

Worked example

Approximate a Gaussian integral on a short interval

Since

e−x2=1−x2+x42!−x63!+⋯ ,e^{-x^2}=1-x^2+\frac{x^4}{2!}-\frac{x^6}{3!}+\cdots,

we have

∫01/2e−x2 dx=[x−x33+x510−x742+⋯ ]01/2.\int_0^{1/2}e^{-x^2}\,dx =\left[x-\frac{x^3}{3}+\frac{x^5}{10}-\frac{x^7}{42}+\cdots\right]_0^{1/2}.

Using the first three integrated terms gives

12−124+1320=0.461458….\frac12-\frac{1}{24}+\frac{1}{320}=0.461458\ldots.

The next integrated term has magnitude 1/(42⋅27)<0.0001871/(42\cdot2^7)<0.000187, which supplies an alternating error bound.

Common mistake

Integrate the coefficients as well as the powers

The term cnxnc_nx^n integrates to cnxn+1/(n+1)c_nx^{n+1}/(n+1). Forgetting the new denominator changes every approximation after the first term.

Interactive checku4b-series_for_definite_integrals-01

Use three terms to approximate ∫00.5e−x2 dx\int_0^{0.5}e^{-x^2}\,dx.

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Show hint

Integrate 1−x2+x4/21-x^2+x^4/2 term by term.

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Exercise

Approximate ∫00.2cos⁡(x2)dx\int_0^{0.2}\cos(x^2)dx.

Exercise

Bound the first omitted term in the Gaussian example.

Exercise

Explain why the method works even without an elementary antiderivative.

Exercise

Compare with Simpson's rule conceptually.

After the explanation

Use the section idea

Reading lens

Pair every Taylor approximation with a degree, center, target input, and certified remainder bound.

Mental model

The polynomial supplies the estimate; the remainder theorem supplies the trust boundary.

Decision

Choose a tractable center and degree, bound the needed derivative on the whole interval, then compare the bound with the required tolerance.

Common trap

Evaluating the next term without checking that the theorem's hypotheses make it a valid error bound.

Check yourself

Can you state exactly why the reported digits are certified?

Source & rights

Original instruction with traceable references.

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