Integrals
We will cover following topics
Riemann Integral
Let \([a,b]\) be a given interval. A partition \(P\) of \([a,b]\) is a finite set of points \(x_0\), \(x_1\), \(x_2\),…\(x_n\) such that \(a=x_{0} \leq x_{1} \leq \cdots \leq x_{n-1} \leq x_{n}=b\) and is denoted by \(P=\left\{x_{0}, x_{1}, x_{2}, \dots, x_{n}\right\}\).
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The upper Riemann sum is given by:
\[U(P, f)=\sum_{1}^{n} M_{i} \Delta x_{i}\]where \(M_{i}=\sup \left\{f(x): x_{i-1} \leq x \leq x_{i}\right\}\) and \(\Delta x_{i}=x_{i}-x_{i-1}\)
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The lower Riemann sum is given by:
\[L(P, f)=\sum_{1}^{n} m_{i} \Delta x_{i}\]where \(m_{i}=\inf \left\{f(x) : x_{i-1} \leq x \leq x_{i}\right\}\) and \(\Delta x_{i}=x_{i}-x_{i-1}\)
Since \(f\) is bounded, \(\exists\) real numbers \(m\) and \(M\) such that \(m \leq f(x) \leq M\) \(\forall\) \(x \in [a,b]\). Thus, for every partition \(P\),
\[m(b-a) \leq L(P, f) \leq U(P, f) \leq M(b-a)\]
We define:
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The upper Riemann integral of \(f\) over \([a,b]\) as \(\int_{a}^{b} f d x=\inf U(P, f)\).
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The lower Riemann integral of \(f\) over \([a,b]\) as \(\int_{a}^{b} f d x=\sup L(P, f)\).
Riemann Integral: \(f\) is said to be Riemann integrable or integrable if the upper and lower integrals of \(f\) are equal and the common value is called the Riemann integral of \(f\), denoted by \(\int_{a}^{b} f d x\).
Improper Integrals
Improper integrals are integrals which are defined for either bounded functions defined on unbounded intervals, or unbounded functions defined over bounded (or unbounded) intervals.
Improper Integral of First Kind
Let \(f\) be Riemann integrable on \([a,x]\) \(\forall\) \(x>a\).
If \(\lim _{x \rightarrow \infty} \int_{a}^{x} f(t) d t=L\), for some \(L \in R\), then we say that the improper integral of first kind, \(\int_{a}^{\infty} f(t) d t\), converges to \(L\), and is denoted by \(\int_{a}^{\infty} f(t) d t=L\).
Otherwise, the improper integral \(\int_{a}^{\infty} f(t) dt\) is said to diverge.
Example: The improper integral \(\int_{1}^{\infty} \dfrac{1}{t^{2}} d t\) converges because \(\int_{1}^{x} \dfrac{1}{t^{2}} d t=1-\dfrac{1}{x} \rightarrow 1 \text { as } x \rightarrow \infty\). On the other hand, the improper integral \(\int_{1}^{\infty} \dfrac{1}{t} d t\) diverges because \(\lim _{x \rightarrow \infty} \int_{1}^{x} \dfrac{1}{t} d t=\lim _{x \rightarrow \infty} \log x \rightarrow \infty\) as \(x \rightarrow \infty\).
We can show that \(\int_{1}^{\infty} \dfrac{1}{t^{p}} d t\) converges to \(\dfrac{1}{p-1}\) for \(p > 1\) and diverges for \(p \leq 1\).
Comparison Test and Limit Comparison Test
Suppose \(f\) is integrable on \([a,x]\) \(\forall\) \(x>a\). Then, we state following two theorems:
Comparison Test: Let \(0 \leq f(t) \leq g(t)\) \(\forall\) \(t>a\). Then,
If \(\int_{a}^{\infty} g(t) d t\) converges, \(\rightarrow \int_{a}^{\infty} f(t) d t\) converges.
Limit Comparison Test (LCT): Let \(f(t)\), \(g(t) \geq 0 \text{ } \forall \text{ } x>a\).
If \(\lim _{t \rightarrow \infty} \dfrac{f(t)}{g(t)}=c\), where \(c \neq 0\), then both the integrals \(\int_{a}^{\infty} f(t) d t\) and \(\int_{a}^{\infty} g(t) d t \text { converge }\) either converge, or diverge together.
If \(c=0\), the convergence of \(\int_{a}^{\infty} g(t) d t\) implies the convergence of \(\int_{a}^{\infty} f(t) d t\).
Example: The integral \(\int_{1}^{\infty} \sin \dfrac{1}{t} d t\) diverges by LCT because \(\dfrac{\sin \dfrac{1}{t}}{\dfrac{1}{t}} \rightarrow 1\) as \(t \rightarrow \infty\). On the other hand, for \(p \in R\), \(\int_{1}^{\infty} e^{-t} t^{p} d t\) converges by LCT because \(\dfrac{e^{-t} t^{p}}{t^{-2}} \rightarrow 0\) as \(x \rightarrow \infty\).
Theorem
If an improper integral \(\int_{a}^{\infty} \vert f(t) \vert d t\) converges, then \(\int_{a}^{\infty} f(t) d t\) converges.
Dirichlet Test
Let \(f, g :[a, \infty) \rightarrow \mathbb{R}\) be such that: (i) \(f\) is decreasing and \(f(t) \rightarrow 0\) as \(t \rightarrow \infty\), (ii) \(g\) is continuous and \(\exists\) \(M\) such that \(\int_{a}^{z} g(t) d t \leq M\) \(\forall\) \(x>a\). Then, \(\int_{a}^{\infty} f(t) g(t) d t\) converges.
Improper Integral of Second Kind:
Let \(\int_{x}^{b} f(t) d t\) exists \(\forall\) \(x\) such that \(a< x \leq b\).
If \(\lim _{x \rightarrow a^{+}} \int_{x}^{b} f(t) d t=M\) for some \(m \in R\), then the improper integral of second kind, \(\int_{a}^{b} f(t) d t\) is said to converge at \(M\) and is denoted by \(\int_{a}^{b} f(t) d t=M\).
Fundamental Theorems Of Integral Calculus
First Fundamental Theorem of Calculus
Let \(f\) be integrable on \([a,b]\). For \(a \leq x \leq b\), let \(F(x)=\int_{a}^{x} f(t) d t\). Then, \(F\) is continuous on \([a,b]\).
Also, if \(f\) is continuous at \(x_0\), then \(F\) is differentiable at \(x_0\) and \(F'(x_0)=(x_0)\).
Second Fundamental Theorem of Calculus
Let \(f\) be integrable on \([a,b]\). If there is a differentiable function \(F\) on \([a,b]\) such that \(F'=f\), then \(\int_{a}^{b} f(x) d x=F(b)-F(a)\).
Riemann Sum
Let \(f :[a, b] \rightarrow \mathbb{R}\) and let \(P=\left\{x_{0}, x_{1}, \ldots, x_{n}\right\}\) be a partition of \([a,b]\).
Let \(c_k \in [x_{k-1},x_k]\), \(k=1\), 2,.., \(n\). Then, corresponding to the partition \(P\) and the intermediate points \(c_k\), a Riemann sum for \(f\) is defined as \(S(P, f)=\sum_{k=1}^{n} f\left(c_{k}\right) \Delta x_{k}\).
Also, the norm of \(P\) is defined as \(\|P\|=\max _{1 \leq i \leq n} \Delta x_{i}\).
PYQs
Riemann Integral
1) Evaluate
\[\int^{\infty}_0 \dfrac{tan^{-1}(ax)}{x(1+x^2)}\,dx\;a>0,\;a\neq1\][2019, 10M]
2) Prove the inequality: \(\dfrac{\pi^2}{9}<\int^{\dfrac{\pi}{2}}_{\dfrac{\pi}{6}}\dfrac{x}{sin x}dx\)<\(\dfrac{2\pi^2}{9}\)
[2018, 10M]
3) Is the function
\(\begin{array}{l} {\qquad f(x)=\left\{\begin{array}{ll}{\frac{1}{n},} & {\frac{1}{n+1}< x \leq \frac{1}{n}} \\ {0,} & {x=0}\end{array}\right.} \end{array}\)
Riemann integrable? If yes, obtain the value of \(\int_{0}^{1} f(x) d x\)
[2015, 15M]
4) Integrate \(\int_{1}^{0} f(x) d x\), where \(f(x)=\left\{\begin{array}{ll}{2 x \sin \dfrac{1}{x} - \cos \dfrac{1}{x}, x} & {\in[0,1]} \\ {0,} & {x=0}\end{array}\right.\)
[2014, 15M]
5) Let \(f(x)=\left\{\begin{array}{ll}{\dfrac{x^{2}}{2}+4} & {\text { if } x \geq 0} \\ {\dfrac{-x^{2}}{2}+2} & {\text { if } x<0}\end{array}\right.\)
Is \(f\) Riemann integrable in the interval \([-1,2]\)? Does there exist a function \(g\) such that \(g^{\prime}(x)=f(x) ?\) Justify your answer.
[2013, 10M]
6) Let \([x]\) denotes the integer part of the real number \(x\), i.e., if \(n \leq x<n+1\) where \(n\) is an integer, then \([x]=n\). Is the function \(f(x)=[x]^{2}+3\) Riemann integrable in the interval \([-1,2]\) If not, explain why. If it is integrable, compute \(\int_{-1}^{2}\left([x]^{2}+3\right) d x\).
[2013, 10M]
7) Give an example of a function \(f(x)\), that is not Riemann integrable but \(\vert f(x) \vert\) is Riemann integrable. Justify your answer.
[2012, 20M]
8) Show that the function \(f(x)\) defined as
\(f(x)=\dfrac{1}{2^{n}}\), \(\dfrac{1}{2^{n+1}} \leq x \leq \dfrac{1}{2^{n}}\), \(n=0, 1,2, \ldots \ldots\) and \(f(0)=0\) is integrable in \([0, 1]\), although it has an infinite number of points of discontinuity. Show that \(\int_{0}^{1} f(x) d x=\dfrac{2}{3}\).
[2004, 12M]
9) A function \(f\) is defined in the interval \((a, b)\) as follows:
\(f(x)=\left \{ \begin{array}{l}{\dfrac{1}{q^{2}} \text { when } x=\dfrac{p}{q}} \\ {\dfrac{1}{q^{3}} \text { when } x=\sqrt{\dfrac{p}{q}}}\end{array}\right.\)
where \(p\), \(q\) are relatively prime integers.
\(f(x)=0\) for all other values of \(x\).
Is \(f\) Riemann integrable? Justify your answer.
[2001, 20M]
Multiple Integrals
1) Find the volume of the solid in the first octant bounded by the paraboloid \(z= 36-4x^2-9y^2\).
[2007, 20M]
2) Find the volume of the ellipsoid
\[\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}+\dfrac{z^2}{c^2}=1\][2006, 20M]
3) Evaluate
\[\iiint ln(x+y+z) dx dy dz\][2005, 12M]
4) Find the volume bounded by the paraboloid \(x^2+y^2=az\), the cylinder \(x^2+y^2=2ay\) and the plane \(z=0\).
[2004, 20M]
5) The axes of two equal cylinders intersect at right angles. If \(a\) be their radius, then find the volume common to cylinders by the method of multiple integrals.
[2003, 20]
6) A solid hemisphere \(H\) of radius \('a'\) has density \(\rho\) depending on the distance \(R\) from the centre and is given by: \(\rho = k (2a-R)\), where \(k\) is a constant.
Find the mass of the hemisphere, by the method of multiple integrals.
[2002, 15M]
7) Evaluate \(\iiint (ax^2+by^2+cz^2)dxdydz\) taken throughout the region \(x^2+y^2+z^2 \leq R^2\).
[2001, 15M]
Improper Integrals
1) Discuss the convergence of \(\int^2_1 \dfrac{\sqrt{x}}{\ln x} dx\).
[2019, 15M]
2) Test the convergence of the improper integral \(\int_{1}^{\infty} \dfrac{d x}{x^{2}\left(1+e^{-x}\right)}\).
[2014, 10M]
3) Examine the convergence of \(\int_{0}^{1} \dfrac{d x}{x^{1 / 2}(1-x)^{1 / 2}}\).
[2006, 12M]
4) Show that \(\int_{0}^{\infty} e^{-t} t^{n-1} d t\) is an improper integral which converges for \(n>0\).
[2005, 30M]
5) Show that \(\int_{0}^{\infty} \dfrac{d x}{1+x^{2} \sin ^{2} x}\) is divergent.
[2003, 20M]
6) Prove that the integral \(\int_{0}^{\infty} x^{m-1} e^{-x} d x\) is convergent if and only if \(m>0\).
[2002, 12M]
7) Show that \(\int_{0}^{\pi / 2} \dfrac{x^{n}}{\sin ^{m} x} d x\) exists if and only if \(m < n+1\).
[2001, 12M]
Fundamental Theorems Of Integral Calculus
1) Let \(f(x)\) be differentiable on \([0,1]\) such that \(f(1)=0\) and \(\int_{0}^{1} f^{2}(x) d x=1\). Prove that \(\int_{0}^{1} x f(x) f^{\prime}(x) d x=-\dfrac{1}{2}\).
[2012, 15M]
2) Let \(f\) be a continuous function on \([0,1]\). Using first Mean Value theorem on Integration, prove that \(\lim _{n \rightarrow \infty} \int_{0}^{1} \dfrac{n f(x)}{1+n^{2} x^{2}} d x=\dfrac{\pi}{2} f(0)\).
[2008, 15M]