Integrals

Let $I=\left[a,b\right]\subset \mathbb{R}$ be a closed bounded interval. A partition of $I$ is a finite set of bounded intervals contained in $I$ such that every $x \in I$ lies in exactly one of the intervals. The intervals may be open, closed or half-open, and may be single points $[x,x]=\{x\}.$ For example, $\{[0,\tfrac{1}{2}),[\tfrac{1}{2},1]\}$ is a partition of $[0,1]$ whereas $\{[0,\tfrac{1}{2}],[\tfrac{1}{2},1]\}$ is not, because $\tfrac{1}{2}$ lies in both intervals. The lengths of the intervals in a partition of $[a,b]$ sum to $b-a.$

Let $I=\left[a,b\right]$ be a closed bounded interval and $P$ and $P'$ be partitions of $I.$ The partition $P'$ is a refinement of $P$ if and only if every interval in $P'$ is contained in some interval in $P.$

Let $I=\left[a,b\right]$ be a closed bounded interval and $P$ and $P'$ be partitions of $I.$ The common refinement of $P$ and $P'$ is the partition formed from the non-empty intersections $J\cap J'$ of intervals $J\in P$ and $J'\in P'.$ The common refinement is a refinement of both $P$ and $P'.$

Let $I=\left[a,b\right]$ be a closed bounded interval, $f:I\rightarrow\mathbb{R}$ be bounded and $P$ be a partition of $I.$ The upper sum of $f$ with respect to $P$ is the sum over the non-empty intervals of $P$ of the supremum of $f$ on each interval weighted by the length of the interval. The lower sum of $f$ with respect to $P$ is the corresponding sum with the infimum in place of the supremum.

$$U(f,P):=\sum_{J\in P,\,J\neq\varnothing}\left(\sup_{x\in J}f(x)\right)\lvert J\rvert$$ $$L(f,P):=\sum_{J\in P,\,J\neq\varnothing}\left(\inf_{x\in J}f(x)\right)\lvert J\rvert$$

Upper and lower sums have the following properties:

• $L(f,P)\leq U(f,P)$ for every partition $P;$
• refining $P$ cannot increase the upper sum or decrease the lower sum;
• by passing to a common refinement, $L(f,P)\leq U(f,P')$ for any two partitions $P$ and $P'.$

Let $I=\left[a,b\right]$ be a closed bounded interval and $f:I\rightarrow\mathbb{R}$ be bounded. The upper integral of $f$ on $I$ is the infimum of the upper sums over all partitions of $I,$ and the lower integral of $f$ on $I$ is the supremum of the lower sums over all partitions of $I.$

$$\overline{\int_a^b}f(x)\,dx:=\inf_P U(f,P),\qquad\underline{\int_a^b}f(x)\,dx:=\sup_P L(f,P)$$

Let $I=\left[a,b\right]$ be a closed bounded interval and $f:I\rightarrow\mathbb{R}$ be bounded. The function $f$ is Riemann integrable on $I$ if and only if its upper and lower integrals are equal. Equivalently, $f$ is Riemann integrable on $I$ if and only if for every $\varepsilon>0$ there exists a partition $P$ with $U(f,P)-L(f,P)\leq\varepsilon.$ The set of Riemann integrable functions on $[a,b]$ is written $\mathcal{R}[a,b].$

Let $I=\left[a,b\right]$ be a closed bounded interval and $f\in\mathcal{R}[a,b].$ The Riemann integral of $f$ over $I$ is the common value of the upper and lower integrals.

$$\int_a^b f(x)\,dx:=\overline{\int_a^b}f(x)\,dx=\underline{\int_a^b}f(x)\,dx$$

The function $f$ is the integrand, the numbers $a$ and $b$ are the limits of integration and $x$ is a dummy variable.

Let $I=\left[a,b\right]$ be a closed bounded interval and $P$ be a partition of $I.$ A tagged partition is the partition $P$ together with a point $\xi_J\in J$ for each non-empty $J\in P.$

Let $I=\left[a,b\right]$ be a closed bounded interval, $f:I\rightarrow\mathbb{R}$ be bounded and $P$ be a tagged partition of $I.$ The Riemann sum of $f$ with respect to $P$ is the sum of the values of $f$ at the tags weighted by the lengths of the intervals.

$$\sum_{J\in P,\,J\neq\varnothing}f\left(\xi_J\right)\lvert J\rvert$$

Every Riemann sum lies between $L(f,P)$ and $U(f,P),$ so if $f\in\mathcal{R}[a,b]$ then the Riemann sums converge to $\int_a^b f(x)\,dx$ as the length of the longest interval in $P$ tends to $0.$

Riemann integrable functions have the following properties:

• if $f:[a,b]\rightarrow\mathbb{R}$ is continuous, or bounded and monotonic, or bounded with finitely many discontinuities, then $f\in\mathcal{R}[a,b];$
• not every bounded function is Riemann integrable: the function on $[0,1]$ equal to $1$ at rational numbers and $0$ at irrational numbers has lower integral $0$ and upper integral $1.$

Let $f,g\in\mathcal{R}[a,b],$ $c\in\mathbb{R}$ and $a<d<b.$ The Riemann integral has the following properties:

$$\int_a^b\left(f(x)+g(x)\right)dx=\int_a^b f(x)\,dx+\int_a^b g(x)\,dx,\qquad\int_a^b c\,f(x)\,dx=c\int_a^b f(x)\,dx$$ $$\mathrm{if}\,\,f(x)\leq g(x)\,\,\forall\,x\in[a,b]\,\,\mathrm{then}\,\,\int_a^b f(x)\,dx\leq\int_a^b g(x)\,dx$$ $$\left\lvert\int_a^b f(x)\,dx\right\rvert\leq\int_a^b\lvert f(x)\rvert\,dx$$ $$\int_a^b f(x)\,dx=\int_a^d f(x)\,dx+\int_d^b f(x)\,dx$$ $$\left(b-a\right)\inf_{x\in[a,b]}f(x)\leq\int_a^b f(x)\,dx\leq\left(b-a\right)\sup_{x\in[a,b]}f(x)$$

The functions $f\,g,$ $\lvert f\rvert$ and $\max\{f,g\}$ are also in $\mathcal{R}[a,b].$ By convention $\int_a^a f(x)\,dx=0$ and $\int_b^a f(x)\,dx=-\int_a^b f(x)\,dx,$ so that the additivity property holds for any ordering of $a,$ $b$ and $d.$

Let $f_n\in\mathcal{R}[a,b]$ be a sequence of functions that converges uniformly to $f$ on $[a,b].$ Then $f\in\mathcal{R}[a,b]$ and the integrals converge.

$$\int_a^b f_n(x)\,dx\rightarrow\int_a^b f(x)\,dx$$

A uniformly convergent series of integrable functions may therefore be integrated term by term. In particular a power series may be integrated term by term over any closed interval inside its interval of convergence.

Let $f:[a,b]\rightarrow\mathbb{R}$ be continuous. The mean value theorem for integrals states that $f$ equals its average value at some point of $[a,b].$

$$\exists\,c\in[a,b] : f(c)=\frac{1}{b-a}\int_a^b f(x)\,dx$$

Let $f\in\mathcal{R}[a,b].$ The integral function of $f$ is the function $F:[a,b]\rightarrow\mathbb{R}$ obtained by integrating $f$ up to a variable upper limit.

$$F(x):=\int_a^x f(t)\,dt$$

Let $f\in\mathcal{R}[a,b]$ with integral function $F.$ The first fundamental theorem of calculus states that $F$ is Lipschitz continuous on $[a,b]$ and that $F$ is differentiable at every $x_0\in[a,b]$ at which $f$ is continuous, with derivative $f(x_0).$

$$F'(x_0)=f(x_0)$$

Let $f:[a,b]\rightarrow\mathbb{R}$ be a function. An antiderivative or primitive of $f$ on $[a,b]$ is a function $F:[a,b]\rightarrow\mathbb{R}$ that is differentiable on $[a,b]$ with $F'=f.$ Antiderivatives have the following properties:

• by the mean value theorem, any two antiderivatives of $f$ on $[a,b]$ differ by a constant;
• by the first fundamental theorem of calculus, every continuous function on $[a,b]$ has an antiderivative.

Let $f:[a,b]\rightarrow\mathbb{R}$ be a function with an antiderivative $F.$ The indefinite integral of $f,$ written $\int f(x)\,dx=F(x)+C,$ is the family of all antiderivatives of $f.$ The number $C\in\mathbb{R}$ is the constant of integration.

Let $f\in\mathcal{R}[a,b]$ and $F$ be an antiderivative of $f$ on $[a,b].$ The second fundamental theorem of calculus states that the integral of $f$ is the net change in $F.$

$$\int_a^b f(x)\,dx=F(b)-F(a)=:\Big[F(x)\Big]_a^b$$

Together the two fundamental theorems state that differentiation and integration are inverse operations, up to a constant. The following common antiderivatives are valid on any interval where the integrand is defined:

$$\int x^n\,dx=\frac{x^{n+1}}{n+1}+C\,\,\left(n\neq -1\right),\qquad\int\frac{1}{x}\,dx=\log_e\lvert x\rvert+C,\qquad\int e^x\,dx=e^x+C$$ $$\int\sin x\,dx=-\cos x+C,\qquad\int\cos x\,dx=\sin x+C$$

Let $f,g\in C^1[a,b].$ Integration by parts is the product rule in integral form.

$$\int_a^b f(x)\,g'(x)\,dx=\Big[f(x)\,g(x)\Big]_a^b-\int_a^b f'(x)\,g(x)\,dx$$

Let $\varphi\in C^1[\alpha,\beta]$ have image $\varphi([\alpha,\beta])\subseteq[a,b]$ and $f$ be continuous on $[a,b].$ Integration by substitution is the chain rule in integral form.

$$\int_{\alpha}^{\beta}f\left(\varphi(t)\right)\varphi'(t)\,dt=\int_{\varphi(\alpha)}^{\varphi(\beta)}f(x)\,dx$$

Let $g:[a,\infty)\rightarrow\mathbb{R}$ be a function and $L\in\mathbb{R}.$ The function $g$ has the limit $L$ as $c\rightarrow\infty,$ written $\lim_{c\rightarrow\infty}g(c)=L,$ if and only if for every $\varepsilon>0$ there exists $M$ such that $\lvert g(c)-L\rvert\leq\varepsilon$ for all $c>M.$

Let $f:[a,\infty)\rightarrow\mathbb{R}$ be a function with $f\in\mathcal{R}[a,c]$ for every $c>a.$ The improper integral of $f$ over $[a,\infty)$ is the limit of the integrals over $[a,c]$ as $c\rightarrow\infty.$

$$\int_a^{\infty}f(x)\,dx:=\lim_{c\rightarrow\infty}\int_a^c f(x)\,dx$$

The improper integral converges if and only if this limit exists, and diverges otherwise. The improper integral $\int_{-\infty}^b f(x)\,dx$ is defined in the same way. The improper integral $\int_{-\infty}^{\infty}f(x)\,dx=\int_{-\infty}^d f(x)\,dx+\int_d^{\infty}f(x)\,dx$ converges if and only if both integrals on the right converge for some, and hence every, $d\in\mathbb{R}.$

Let $f:(a,b]\rightarrow\mathbb{R}$ be a function with $f\in\mathcal{R}[c,b]$ for every $c\in(a,b),$ which allows $f$ to be unbounded near $a.$ The improper integral of $f$ over $(a,b]$ is the limit of the integrals over $[c,b]$ as $c$ decreases to $a.$

$$\int_a^b f(x)\,dx:=\lim_{c\rightarrow a,\,c>a}\int_c^b f(x)\,dx$$

The improper integral of a function unbounded near $b$ is defined in the same way. For example,

$$\int_1^{\infty}\frac{1}{x^p}\,dx=\frac{1}{p-1}\,\,\mathrm{converges}\,\,\mathrm{if}\,\mathrm{and}\,\mathrm{only}\,\mathrm{if}\,\,p>1$$ $$\int_0^1\frac{1}{x^p}\,dx=\frac{1}{1-p}\,\,\mathrm{converges}\,\,\mathrm{if}\,\mathrm{and}\,\mathrm{only}\,\mathrm{if}\,\,p<1.$$

Let $\int f(x)\,dx$ be an improper integral. The improper integral is absolutely convergent if and only if $\int\lvert f(x)\rvert\,dx$ converges. An absolutely convergent improper integral converges.

Let $f$ and $g$ be functions on the same range of integration. The comparison test for integrals states that if $\lvert f(x)\rvert\leq g(x)$ on the range of integration and $\int g(x)\,dx$ converges then $\int f(x)\,dx$ is absolutely convergent.

The Gaussian integral is the improper integral of $e^{-x^2}$ over $\mathbb{R}.$

$$\int_{-\infty}^{\infty}e^{-x^2}\,dx=\sqrt{\pi}$$

A rectangle in $\mathbb{R}^n$ is a Cartesian product $R=\left[a_1,b_1\right]\times\cdots\times\left[a_n,b_n\right]$ of $n$ closed bounded intervals. The volume of $R$ is the product of the lengths of its sides.

$$\lvert R\rvert:=\prod_{i=1}^{n}\left(b_i-a_i\right)$$

Let $R=\left[a_1,b_1\right]\times\cdots\times\left[a_n,b_n\right]$ be a rectangle. A grid partition of $R$ is a set of rectangles of the form $J_1\times\cdots\times J_n$ where $J_i$ ranges over the intervals of a partition of $\left[a_i,b_i\right]$ for each $i.$

Let $R\subset\mathbb{R}^n$ be a rectangle and $f:R\rightarrow\mathbb{R}$ be bounded. The upper and lower sums of $f$ with respect to a grid partition, the upper and lower integrals of $f$ and the multiple integral of $f$ over $R,$ written $\int_R f(\underline{x})\,d\underline{x},$ are defined exactly as in the univariate case with lengths of intervals replaced by volumes of rectangles. The multiple integral has the following properties:

• continuous functions on $R$ are integrable;
• the multiple integral is linear and monotonic;
• the multiple integral is additive over rectangles that overlap only on their boundaries.

Let $D\subseteq\mathbb{R}^n.$ The indicator function of $D$ is the function $\mathbf{1}_D:\mathbb{R}^n\rightarrow\{0,1\}$ that is $1$ on $D$ and $0$ elsewhere.

Let $D\subseteq\mathbb{R}^n$ be a bounded set, $R$ be a rectangle containing $D$ and $f:D\rightarrow\mathbb{R}$ be a function. The integral of $f$ over $D$ is the integral over $R$ of $f$ multiplied by the indicator function of $D,$ whenever this integral exists.

$$\int_D f(\underline{x})\,d\underline{x}:=\int_R f(\underline{x})\,\mathbf{1}_D(\underline{x})\,d\underline{x}$$

The volume of $D$ is $\int_D 1\,d\underline{x}.$

Let $R=[a,b]\times[c,d]$ be a rectangle in $\mathbb{R}^2$ and $f:R\rightarrow\mathbb{R}$ be continuous. The iterated integrals of $f$ over $R$ are the integrals obtained by integrating one variable at a time in either order. Fubini’s theorem states that the multiple integral of $f$ equals both iterated integrals.

$$\int_R f(x,y)\,d(x,y)=\int_a^b\left(\int_c^d f(x,y)\,dy\right)dx=\int_c^d\left(\int_a^b f(x,y)\,dx\right)dy$$

Fubini’s theorem extends to continuous functions on rectangles in $\mathbb{R}^n,$ which can be integrated one coordinate at a time in any order, and to integrable functions whose iterated integrals exist. On a non-rectangular region the limits of the inner integral depend on the outer variable. For example, if $D=\{(x,y):a\leq x\leq b,\,g_1(x)\leq y\leq g_2(x)\}$ for continuous $g_1\leq g_2$ then $\int_D f(x,y)\,d(x,y)=\int_a^b\int_{g_1(x)}^{g_2(x)}f(x,y)\,dy\,dx.$

Let $U,V\subseteq\mathbb{R}^n$ be open sets, $\underline{\varphi}:U\rightarrow V$ be a bijection such that $\underline{\varphi}$ and $\underline{\varphi}^{-1}$ are both continuously differentiable and $f:V\rightarrow\mathbb{R}$ be integrable. The change of variables theorem states that

$$\int_V f(\underline{y})\,d\underline{y}=\int_U f\left(\underline{\varphi}(\underline{x})\right)\left\lvert\det J_{\underline{\varphi}}(\underline{x})\right\rvert\,d\underline{x}.$$

The factor $\left\lvert\det J_{\underline{\varphi}}(\underline{x})\right\rvert,$ the absolute value of the determinant of the Jacobian matrix of $\underline{\varphi},$ is the factor by which $\underline{\varphi}$ scales volume near $\underline{x}.$ In two dimensions this factor is

$$\left\lvert\det J_{\underline{\varphi}}(\underline{x})\right\rvert=\left\lvert\frac{\partial\varphi_1}{\partial x_1}\frac{\partial\varphi_2}{\partial x_2}-\frac{\partial\varphi_1}{\partial x_2}\frac{\partial\varphi_2}{\partial x_1}\right\rvert.$$

Polar coordinates are the change of variables $\underline{\varphi}(r,\theta)=\left(r\cos\theta,r\sin\theta\right)$ on $\mathbb{R}^2,$ which has $\left\lvert\det J_{\underline{\varphi}}(r,\theta)\right\rvert=r,$ so that

$$\int_{\mathbb{R}^2}f(x,y)\,d(x,y)=\int_0^{2\pi}\int_0^{\infty}f\left(r\cos\theta,r\sin\theta\right)r\,dr\,d\theta.$$

Applied to $f(x,y)=e^{-x^2-y^2},$ polar coordinates evaluate the Gaussian integral.