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Limits of integration

Limits of integration are the specified lower and upper bounds that define the interval or region over which a definite integral is evaluated, determining the extent of the function's domain in the integration process.^[1] In the standard notation for a single-variable definite integral, \int_a^b f(x) \, dx, the value a represents the lower limit and b the upper limit, with the integral computing the net signed area under the curve of f(x) from x = a to x = b.^[2] These limits are crucial for applying the Fundamental Theorem of Calculus, which evaluates the integral as F(b) - F(a), where F is an antiderivative of f.^[3] In multivariable calculus, limits of integration extend to multiple integrals, where they outline the boundaries of a region in higher dimensions, often depending on other variables to describe non-rectangular domains.^[4] For double integrals over Type I regions (vertically simple), the inner limits for y range from a lower function g_1(x) to an upper function g_2(x), while the outer limits for x are constants from a to b; conversely, Type II regions (horizontally simple) reverse this, with inner limits for x as functions of y and outer limits constants for y.^[4] This setup allows computation of volumes, masses, or other quantities over irregular shapes by iterated integration.^[5] Key properties of limits of integration include the reversal rule, where interchanging the upper and lower limits negates the integral's value: \int_a^b f(x) \, dx = -\int_b^a f(x) \, dx, and the zero integral property when limits coincide: \int_a^a f(x) \, dx = 0.^[2] For improper integrals, limits may extend to infinity, requiring evaluation as limits of proper integrals to check convergence.^[6] Correctly determining limits, especially in multiple integrals, often involves sketching the region to ensure accurate bounds and order of integration.^[7]

Definite Integrals

Definition and Notation

In calculus, the limits of integration refer to the endpoints that define the interval over which a definite integral is computed. The definite integral \int_a^b f(x) \, dx, where a is the lower limit and b is the upper limit with a \leq b, represents the net signed area between the curve of the function f(x) and the x-axis from x = a to x = b. This area is positive where f(x) is above the x-axis and negative where it is below, with the limits specifying the precise interval of accumulation./Chapter_5:_Integration/5.2:_The_Definite_Integral) The notation for the definite integral arises from the limit definition using Riemann sums, which approximate the area by partitioning the interval [a, b] into n subintervals of width \Delta x = (b - a)/n and summing rectangles with heights f(x_i^*) at sample points x_i^* in each subinterval. As the number of subintervals increases and the maximum width approaches zero, the Riemann sum converges to the integral:

\int_a^b f(x) \, dx = \lim_{n \to \infty} \sum_{i=1}^n f(x_i^*) \Delta x.

This limit process formalizes the definite integral, with the bounds a and b determining the domain of summation.^[8] Geometrically, the limits of integration delineate the region whose signed areas are accumulated; for instance, changing a or b alters the net area by including or excluding portions of the curve. The concept of limits as bounds in integration was developed in the 17th century by Isaac Newton and Gottfried Wilhelm Leibniz, who independently laid the foundations of calculus, treating integrals as sums over infinitesimal intervals.^[9] A simple example illustrates the role of these limits: the integral \int_0^1 x \, dx = \frac{1}{2} computes the net area under the line y = x from x = 0 to x = 1, a triangular region with base 1 and height 1; shifting the lower limit to x = 0.5 would yield \int_{0.5}^1 x \, dx = \frac{3}{8}, demonstrating how the bounds control the extent of integration./Chapter_5:_Integration/5.2:_The_Definite_Integral) The definite integral relates to antiderivatives via the Fundamental Theorem of Calculus.^[8]

Properties and Evaluation

The properties of definite integrals with fixed finite limits a and b (where a < b) enable efficient manipulation and evaluation, assuming the integrand is continuous on [a, b]. These properties stem from the definition of the definite integral as a limit of Riemann sums and facilitate computations without resorting to the original summation process./05%3A_Integration/5.02%3A_The_Definite_Integral) A cornerstone for evaluation is the Fundamental Theorem of Calculus (Part 1), which states that if f is continuous on [a, b] and F is an antiderivative of f (i.e., F'(x) = f(x)), then

\int_a^b f(x) \, dx = F(b) - F(a).

This theorem directly substitutes the fixed limits into the antiderivative, transforming the integral into a difference of function values at the endpoints.^[10] Linearity properties allow decomposition of integrals. For constants c and functions f, g continuous on [a, b],

\int_a^b [c f(x) + g(x)] \, dx = c \int_a^b f(x) \, dx + \int_a^b g(x) \, dx.

Additionally, additivity over subintervals holds: for a < c < b,

\int_a^b f(x) \, dx = \int_a^c f(x) \, dx + \int_c^b f(x) \, dx.

These rules preserve the fixed limits while enabling simplification of complex integrands./05%3A_Integration/5.02%3A_The_Definite_Integral) The directional role of limits is evident in the reversal property:

\int_a^b f(x) \, dx = -\int_b^a f(x) \, dx.

Swapping the limits introduces a negative sign, reflecting the orientation of the interval in the integral's signed area interpretation.^[11] To evaluate using these properties, consider \int_1^4 (2x + 1) \, dx. The antiderivative is F(x) = x^2 + x, so by the Fundamental Theorem,

\int_1^4 (2x + 1) \, dx = F(4) - F(1) = (16 + 4) - (1 + 1) = 20 - 2 = 18.

This step-by-step substitution of limits yields the net area under the line from x=1 to x=4.^[10] For continuous functions on [a, b], the Mean Value Theorem for Integrals guarantees the existence of some c \in [a, b] such that

\int_a^b f(x) \, dx = f(c) (b - a),

where f(c) represents the average value of f over the interval. This theorem connects the integral to pointwise function values, aiding in bounds and approximations.^[12]

Changing Limits in Integration Techniques

Substitution (U-Substitution)

In the substitution method, known as u-substitution, a change of variables is used to simplify the evaluation of definite integrals of the form \int_a^b f(g(x)) g'(x) \, dx. By setting u = g(x), the differential becomes du = g'(x) \, dx, transforming the integral into \int_{g(a)}^{g(b)} f(u) \, du. This adjustment of the limits of integration—from the original interval [a, b] in x to the corresponding interval [g(a), g(b)] in u—ensures that the value of the definite integral remains unchanged under the substitution, preserving the area represented by the integral.^[13] The process involves several key steps to correctly apply u-substitution to definite integrals. First, identify a suitable substitution u = g(x) such that du = g'(x) \, dx matches part of the integrand. Next, determine the new limits by evaluating u at the original bounds: the lower limit becomes u(a) = g(a) and the upper limit u(b) = g(b). Then, rewrite the integral entirely in terms of u, using the new limits, and evaluate it as a standard definite integral in u. This method leverages the chain rule in reverse, aligning with the fundamental theorem of calculus for definite integrals.^[13]^[14] Consider the integral \int_1^e \frac{\ln x}{x} \, dx. Let u = \ln x, so du = \frac{1}{x} \, dx. The limits change as follows: when x = 1, u = 0; when x = e, u = 1. The integral becomes \int_0^1 u \, du = \left[ \frac{u^2}{2} \right]_0^1 = \frac{1}{2}. This example illustrates how substitution simplifies logarithmic forms while correctly transforming the interval.^[13] Another representative case is \int_0^{\pi/2} \sin^2 x \cos x \, dx. Set u = \sin x, then du = \cos x \, dx. The bounds shift to u(0) = 0 and u(\pi/2) = 1. Substituting yields \int_0^1 u^2 \, du = \left[ \frac{u^3}{3} \right]_0^1 = \frac{1}{3}. Here, the technique handles trigonometric powers efficiently by aligning the differential with the integrand.^[13] A frequent error in applying u-substitution to definite integrals is neglecting to adjust the limits, which effectively treats the integral as indefinite and leads to incorrect results upon evaluation. For instance, using the original limits after substitution would mismatch the variable domains, altering the computed value. Always verify the limit transformation to avoid this pitfall, as emphasized in standard calculus procedures.^[14]^[15]

Integration by Parts

Integration by parts is a technique for evaluating definite integrals of products of functions, derived from the product rule for differentiation applied to the fundamental theorem of calculus. The formula for a definite integral from a to b is

\int_a^b u \, dv = \left[ u v \right]_a^b - \int_a^b v \, du,

where the boundary term \left[ u v \right]_a^b = u(b) v(b) - u(a) v(a) captures the contributions from the limits, and the remaining integral over v \, du retains the same limits a to b.^[16] To apply this, select functions u and dv such that du is simpler than u and v = \int dv is computable, then substitute into the formula: evaluate u v at the upper limit b minus at the lower limit a, and subtract the definite integral of v \, du from a to b. This process isolates the boundary effects explicitly, distinguishing it from the indefinite case where no limits are evaluated and only an antiderivative plus constant results.^[16]^[17] For example, consider \int_0^1 x e^x \, dx. Let u = x, so du = dx, and dv = e^x \, dx, so v = e^x. Then,

\int_0^1 x e^x \, dx = \left[ x e^x \right]_0^1 - \int_0^1 e^x \, dx = (1 \cdot e^1 - 0 \cdot e^0) - \left[ e^x \right]_0^1 = e - (e - 1) = 1.

The boundary term e arises from the limits, while the subtracted integral simplifies due to the choice of parts.^[16] Another representative case is \int_1^2 \ln x \, dx. Set u = \ln x, so du = \frac{1}{x} dx, and dv = dx, so v = x. Applying the formula yields

\int_1^2 \ln x \, dx = \left[ x \ln x \right]_1^2 - \int_1^2 x \cdot \frac{1}{x} \, dx = \left[ x \ln x \right]_1^2 - \int_1^2 1 \, dx = (2 \ln 2 - 1 \ln 1) - _1^2 = (2 \ln 2 - 0) - (2 - 1) = 2 \ln 2 - 1,

with the boundary evaluation handling the logarithmic behavior at the limits.^[18] For integrals requiring repeated applications of integration by parts, such as products of polynomials and exponentials or trigonometrics, the tabular method organizes the process by tabulating successive derivatives of u and integrals of dv, alternating signs, and evaluating the resulting sum and remainder integral over the limits a to b at each step to incorporate boundary terms efficiently.^[19]

Improper Integrals

Infinite Limits of Integration

Infinite limits of integration arise in improper integrals of Type I, where at least one endpoint of the interval is infinite, such as \int_a^\infty f(x) \, dx, \int_{-\infty}^b f(x) \, dx, or \int_{-\infty}^\infty f(x) \, dx. These are defined as limits of definite integrals over finite intervals that approach the infinite boundary: specifically, \int_a^\infty f(x) \, dx = \lim_{b \to \infty} \int_a^b f(x) \, dx, provided the limit exists.^[20] Similarly, for the lower limit, \int_{-\infty}^b f(x) \, dx = \lim_{a \to -\infty} \int_a^b f(x) \, dx, and for both infinite, the integral over (-\infty, \infty) is handled by splitting at a finite point c and taking the limit as the endpoints approach \pm \infty independently, converging only if both parts do. An improper integral with infinite limits converges if the corresponding limit exists and is finite; otherwise, it diverges. Absolute convergence occurs when \int_a^\infty |f(x)| \, dx converges, implying the original integral converges, though the converse does not hold. For positive integrands, convergence is determined solely by the finiteness of the limit. A classic example of convergence is \int_1^\infty \frac{1}{x^2} \, dx = \lim_{b \to \infty} \left[ -\frac{1}{x} \right]_1^b = \lim_{b \to \infty} \left( -\frac{1}{b} + 1 \right) = 1, which is finite. In contrast, \int_1^\infty \frac{1}{x} \, dx = \lim_{b \to \infty} [\ln x]_1^b = \lim_{b \to \infty} (\ln b - \ln 1) = \infty, so it diverges.^[20] To assess convergence without direct evaluation, the comparison test is useful: if $0 \leq f(x) \leq g(x) for x \geq a and \int_a^\infty g(x) \, dx converges, then \int_a^\infty f(x) \, dx converges; if \int_a^\infty f(x) \, dx diverges, so does \int_a^\infty g(x) \, dx. A special case is the p-integral \int_1^\infty x^{-p} \, dx, which converges if and only if p > 1, providing a benchmark for similar functions via comparison. For instance, since \int_1^\infty x^{-p} \, dx diverges for p \leq 1, any larger positive function will also diverge.^[21]^[22] The development of improper integrals with infinite limits in the 18th century paralleled Leonhard Euler's foundational work on infinite series and the gamma function, which he expressed as an integral from 0 to \infty. Euler's Institutionum calculi integralis (1768–1770) advanced techniques for evaluating such expressions, influencing the rigorous treatment of convergence later formalized by Cauchy and others.

Discontinuous Integrands

Improper integrals with discontinuous integrands, also known as Type II improper integrals, address situations where the integrand f(x) is continuous on a finite interval [a, b] except at one or more points of discontinuity within the interval or at the endpoints. These integrals are evaluated using limits to approach the points of discontinuity, ensuring the process handles the singularity appropriately.^[20] When the discontinuity occurs at an interior point c where a < c < b, the improper integral \int_a^b f(x) \, dx is defined as the limit

\lim_{\epsilon \to 0^+} \left( \int_a^{c - \epsilon} f(x) \, dx + \int_{c + \epsilon}^b f(x) \, dx \right),

provided this limit exists and is finite. For a discontinuity at the lower endpoint a, the integral is expressed as \lim_{\epsilon \to 0^+} \int_{a + \epsilon}^b f(x) \, dx. Similarly, at the upper endpoint b, it is \lim_{\epsilon \to 0^+} \int_a^{b - \epsilon} f(x) \, dx. The integral converges if the limit is finite; otherwise, it diverges. Both subintegrals must converge individually in the case of an interior discontinuity.^[23]^[20] A classic example of convergence involves the integrand with a singularity at the endpoint: consider \int_0^1 x^{-1/2} \, dx. This is evaluated as

\lim_{\epsilon \to 0^+} \int_\epsilon^1 x^{-1/2} \, dx = \lim_{\epsilon \to 0^+} \left[ 2x^{1/2} \right]_\epsilon^1 = \lim_{\epsilon \to 0^+} \left( 2 - 2\epsilon^{1/2} \right) = 2.

The finite value indicates convergence, despite the vertical asymptote at x = 0. In contrast, \int_0^1 \frac{1}{x} \, dx diverges:

\lim_{\epsilon \to 0^+} \int_\epsilon^1 \frac{1}{x} \, dx = \lim_{\epsilon \to 0^+} \left[ \ln x \right]_\epsilon^1 = \lim_{\epsilon \to 0^+} \left( 0 - \ln \epsilon \right) = \infty.

This divergence arises because the antiderivative grows without bound near the singularity.^[24]^[25] For symmetric singularities where the standard improper integral diverges, the Cauchy principal value offers an alternative interpretation by approaching the discontinuity symmetrically from both sides. For instance, \int_{-1}^1 \frac{1}{x} \, dx has a discontinuity at x = 0, and the individual one-sided integrals diverge. However, the principal value is

\lim_{\epsilon \to 0^+} \left( \int_{-1}^{-\epsilon} \frac{1}{x} \, dx + \int_{\epsilon}^1 \frac{1}{x} \, dx \right) = \lim_{\epsilon \to 0^+} \left( \ln \epsilon - \ln 1 + \ln 1 - \ln \epsilon \right) = 0.

This assigns a finite value of 0 to the integral, useful in contexts requiring symmetric treatment of the singularity, though it does not imply convergence in the standard sense.^[26]

References

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The definite integral generalizes the area under a curve, and is defined as a limit of a function on an interval, and is a number, not a family of functions.
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Improper integrals are definite integrals with infinite limits, where the limit exists and has finite values, the integral converges; otherwise, it diverges.
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The Riemann integral is the definite integral normally encountered in calculus texts and used by physicists and engineers.
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The method of integration by parts will transform this integral (left-hand side of equation) into the difference of the product of two functions and a new `` ...
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∫ u dv = uv − ∫ v du. ∫ u dv = uv − ∫ v du. The advantage of using the integration-by-parts formula is that we can use it to exchange one integral for another, ...
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