Integrals
We will cover following topics
Definite Integrals
Riemanns’ Definition of Definite Integrals
The definite integral of a continuous function \(f\) over the interval \([a,b]\) is the limit of a Riemann sum as the number of subdivisions approach infinity. Mathematically,
\[\int_{a}^{b} f(x) d x=\lim _{n \rightarrow \infty} \sum_{i=1}^{n} \Delta x \cdot f\left(x_{i}\right)\]where \(\Delta x=\dfrac{b-a}{n}\), and \(x_{i}=a+\Delta x \cdot i\)
Indefinite Integrals
An integral of the form \(\int f(z) d z\), without lower and upper limits, is called an indefinite integral.
It is also called an antiderivative.
Improper Integrals
If the interval over which the definite integral \(\int_{a}^{b} f(x) d x\) is defined is infinite or if the function \(f(x)\) has a discontinuity in \([a,b]\) or both of these conditions are satisfied, the integral is called an improper integral.
Observations:
-
The improper integral \(\int_{1}^{\infty} \dfrac{1}{x^{p}} d x\) converges when \(p>1\) and diverges when \(p \leq 1\)$.
-
The improper integral \(\int_{0}^{1} \dfrac{1}{x^{p}} d x\) converges when \(p< 1\) and diverges when \(p \geq 1\).
Direct Comparison Test for Improper Integrals: Let \(f\) and \(g\) be continuous on \([a, \infty)\) where \(0 \leq f(x) \leq g(x)\) for all \(x\) in \([a, \infty)\).
- If \(\int_{a}^{\infty} g(x) d x\) converges, then \(\int_{a}^{\infty} f(x) d x\) converges.
- If \(\int_{a}^{\infty} f(x) d x\) diverges, then \(\int_{a}^{\infty} g(x) d x\) diverges.
Limit Comparison Test for Improper Integrals: Let \(f\) and \(g\) be continuous functions on \([a, \infty)\) where \(f(x)>0\) and \(g(x)>0\) for all \(x\). If
\[\lim_{x \rightarrow \infty} \dfrac{f(x)}{g(x)}=L, \text{ } 0< L < \infty\]then \(\int_{a}^{\infty} f(x) d x\) and \(\int_{a}^{\infty} g(x) d x\) either both converge or both diverge.
Double Integrals
Consider a function of two variables \(f(x,y)\). The definite integral denoted by \(\iint_{R} f(x, y) d A\), where \(R\) is the region of integration in \(xy-plane\), is called as a double inetegral.
For positive \(f(x,y)\), this definite integral gives the volume under the surface \(f(x,y)\) and above \(xy-plane\), for \(x\) and \(y\) in the region \(R\).
Triple Integrals
The integration of a function \(f(x,y,z)\) over a three-dimensional region \(R\) in \(xyz-space\) is called a triple integral and is denoted as:
\[\iiint_{R} f(x, y, z) d V\]PYQs
Definite Integrals
1) Evaluate the following integral:
\[\int_{\pi / 6}^{\pi / 3} \dfrac{\sqrt[3]{\sin x}}{\sqrt[3]{\sin x}+\sqrt[3]{\cos x}} d x\][2015, 10M]
2) Evaluate \(\int_{0}^{1} \dfrac{\log _{e}(1+x)}{1+x^{2}} dx\)
[2014, 10M]
Infinite Integrals
1) Show that \(e^{-x} x^{n}\) is bounded on \([0, \infty)\) for all positive integral values of \(n\). Using this result, show that \(\int_{0}^{\infty} e^{-x} x^{n} d x\) exists.
[2007, 25M]
2) Show that \(\int_{0}^{\infty} \int_{0}^{\infty} e^{-\left(x^{2}+y^{2}\right)} d x d y=\dfrac{\pi}{4}\)
[2002, 12M]
Improper Integrals
1) Examine if the improper integral \(\int_{0}^{3} \dfrac{2 x d x}{\left(1-x^{2}\right)^{2 / 3}}\) exists.
[2017, 10M]
2) Evaluate \(I=\int_{0}^{1} \sqrt[3]{x \log \left(\dfrac{1}{x}\right)} d x\)
[2016, 10M]
3) Evaluate \(\int_{0}^{1}\left(2 x \sin \dfrac{1}{x}-\cos \dfrac{1}{x} \right)d x\)
[2013, 10M]
4) Find all the real values of \(p\) and \(q\) so that the integral \(\int_{0}^{1} x^{p}\left(\log \dfrac{1}{x}\right)^{q} d x\) converges.
[2012, 20M]
5) Does the integral \(\int_{-1}^{1} \sqrt{\dfrac{1+x}{1-x}} dx\) exist? If so, find its value.
[2010, 12M]
6) Evaluate \(\int_{0}^{1}(x \ell n x)^{3} d x\)
[2008, 12M]
7) Prove that the function \(f\) defined on \([0,4]\) by \(f(x)=[x]\), the greatest integer \(\leq x, x \in[0,4]\) is integrable on \([0,4]\) and that \(\int_{0}^{4}f(x) d x=6\).
[2004, 12M]
8) Test the convergent of the integrals:-
i) \(\int_{0}^{1} \dfrac{dx}{x^{1/3}\left(1+x^{2}\right)}\)
ii) \(\int_{0}^{\infty} \dfrac{\sin ^{2} x}{x^{2}} d x\)
[2003, 15M]
9) Test the convergence of \(\int_{0}^{1} \dfrac{\sin \left(\dfrac{1}{x}\right)}{\sqrt{x}} d x\).
[2001, 12M]
Double Integrals
1) Evaluate the integral \(\int^a_0 \int^x_{x/a} \dfrac{xdydx}{x^2+y^2}\).
[2018, 12M]
2) Integrate the function \(f(x, y)=x y\left(x^{2}+y^{2}\right)\) over the domain \(R:\left\{-3 \leq x^{2}-y^{2} \leq 3,1 \leq x y \leq 4\right\}\)
[2017, 10M]
3) Prove that \(\dfrac{\pi}{3} \leq \iint \dfrac{d x d y}{\sqrt{x^{2}+(y-2)^{2}}} \leq \pi\), where \(D\) is the unit disc.
[2017, 10M]
4) Evaluate \(\iint_{R} f(x, y) d x d y\) over the rectangle \(R=[0,1 ; 0,1]\), where \(f(x, y)=\left\{\begin{array}{c}{x+y, \text { if } x^{2}<y<2 x^{2}} \\ {0,}\end{array}\right.\)
[2016, 15M]
5) Evaluate the integral \(\iint(x-y)^{2} \cos ^{2}(x+y) d x d y\) where \(R\) is the rhombus with successive vertices as \((\pi, 0)\), \((2 \pi, \pi)\), \((\pi, 2 \pi)\), \((0, \pi)\).
[2015, 12M]
6) Evaluate \(\iint_{R} \sqrt{\vert y-x^{2} \vert} d x d y\) where \(R=[-1,1 ; 0,2]\).
[2015, 13M]
7) Evaluate \(\iint_{D} x y d A\), where \(D\) is the region bounded by the line \(y=x-1\) and the parabola \(y^{2}=2 x+6\).
[2013, 15M]
8) Show that the function
\[f(x) = [ x^2 ] + \vert x-1 \vert\]is Riemann integrable m the interval [0, 2], where \([\alpha]\) denotes the greatest integer less than or equal to \(\alpha\). Can you give an example of a function that is not Riemann integrable on [0, 2]? Compute \(\int_0^2 f(x) dx\), where \(f(x)\) is as above.
[2010, 12M]
9) Evaluate \(\int_{0}^{1} \ln x d x\).
[2011, 12M]
10) Evaluate \(I=\iint_{s} (x d y d z+d z d x+x z^{2}) dx dy\), where \(S\) is the outer side of the part of the sphere \(x^{2}+y^{2}+z^{2}=1\) in the first octant.
[2009, 20M]
11) Evaluate the double integral \(\iint_{y}^{a} \dfrac{x d x d y}{x^{2}+y^{2}}\) by changing the order of integration.
[2008, 20M]
12) Change the order of integration in \(\int_0^{\infty} \int_{x}^{\infty} \dfrac{e^{-y}}{y} d y d x\) and hence evaluate it.
[2006, 15M]
13) Find the x-coordinate of the centre of gravity of the solid lying inside the cylinder \(x^2+y^2=2ax\), between the plane \(z=0\) and the paraboloid \(x^2+y^2=az\).
[2005, 15M]
14) Find the centre of gravity of the region bounded by the curve \(\left(\dfrac{x}{a}\right)^{2/3} + \left(\dfrac{x}{b}\right)^{2/3} =1\) and both axes in the first quadrant, the density being \(\rho =kxy\), where \(k\) is a constant.
[2002, 15M]
Triple Integrals
1) Let \(D\) be the region determine by the inequalities \(x>0, y>0, z< 8\) and \(z>x^{2}+y^{2}\). Compute \(\iiint_{D} 2 x d x d y d z\).
[2010, 20M]
2) Evaluate \(\iiint(x+y+z+1)^{2} d x d y d z\) over the region defined by \(x \geq 0\), \(y \geq 0\), \(z \geq 0\), \(x+y+z \leq 1\).
[2001, 15M]