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,
∫baf(x)dx=limn→∞n∑i=1Δx⋅f(xi)where Δx=b−an, and xi=a+Δx⋅i
Indefinite Integrals
An integral of the form ∫f(z)dz, 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 ∫baf(x)dx 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 ∫∞11xpdx converges when p>1 and diverges when p≤1$.
-
The improper integral ∫101xpdx converges when p<1 and diverges when p≥1.
Direct Comparison Test for Improper Integrals: Let f and g be continuous on [a,∞) where 0≤f(x)≤g(x) for all x in [a,∞).
- If ∫∞ag(x)dx converges, then ∫∞af(x)dx converges.
- If ∫∞af(x)dx diverges, then ∫∞ag(x)dx diverges.
Limit Comparison Test for Improper Integrals: Let f and g be continuous functions on [a,∞) where f(x)>0 and g(x)>0 for all x. If
limx→∞f(x)g(x)=L, 0<L<∞then ∫∞af(x)dx and ∫∞ag(x)dx either both converge or both diverge.
Double Integrals
Consider a function of two variables f(x,y). The definite integral denoted by ∬Rf(x,y)dA, 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:
∭Rf(x,y,z)dVPYQs
Definite Integrals
1) Evaluate the following integral:
∫π/3π/63√sinx3√sinx+3√cosxdx[2015, 10M]
2) Evaluate ∫10loge(1+x)1+x2dx
[2014, 10M]
Infinite Integrals
1) Show that e−xxn is bounded on [0,∞) for all positive integral values of n. Using this result, show that ∫∞0e−xxndx exists.
[2007, 25M]
2) Show that ∫∞0∫∞0e−(x2+y2)dxdy=π4
[2002, 12M]
Improper Integrals
1) Examine if the improper integral ∫302xdx(1−x2)2/3 exists.
[2017, 10M]
2) Evaluate I=∫103√xlog(1x)dx
[2016, 10M]
3) Evaluate ∫10(2xsin1x−cos1x)dx
[2013, 10M]
4) Find all the real values of p and q so that the integral ∫10xp(log1x)qdx converges.
[2012, 20M]
5) Does the integral ∫1−1√1+x1−xdx exist? If so, find its value.
[2010, 12M]
6) Evaluate ∫10(xℓnx)3dx
[2008, 12M]
7) Prove that the function f defined on [0,4] by f(x)=[x], the greatest integer ≤x,x∈[0,4] is integrable on [0,4] and that ∫40f(x)dx=6.
[2004, 12M]
8) Test the convergent of the integrals:-
i) ∫10dxx1/3(1+x2)
ii) ∫∞0sin2xx2dx
[2003, 15M]
9) Test the convergence of ∫10sin(1x)√xdx.
[2001, 12M]
Double Integrals
1) Evaluate the integral ∫a0∫xx/axdydxx2+y2.
[2018, 12M]
2) Integrate the function f(x,y)=xy(x2+y2) over the domain R:{−3≤x2−y2≤3,1≤xy≤4}
[2017, 10M]
3) Prove that π3≤∬dxdy√x2+(y−2)2≤π, where D is the unit disc.
[2017, 10M]
4) Evaluate ∬Rf(x,y)dxdy over the rectangle R=[0,1;0,1], where f(x,y)={x+y, if x2<y<2x20,
[2016, 15M]
5) Evaluate the integral ∬(x−y)2cos2(x+y)dxdy where R is the rhombus with successive vertices as (π,0), (2π,π), (π,2π), (0,π).
[2015, 12M]
6) Evaluate ∬R√|y−x2|dxdy where R=[−1,1;0,2].
[2015, 13M]
7) Evaluate ∬DxydA, where D is the region bounded by the line y=x−1 and the parabola y2=2x+6.
[2013, 15M]
8) Show that the function
f(x)=[x2]+|x−1|is Riemann integrable m the interval [0, 2], where [α] denotes the greatest integer less than or equal to α. Can you give an example of a function that is not Riemann integrable on [0, 2]? Compute ∫20f(x)dx, where f(x) is as above.
[2010, 12M]
9) Evaluate ∫10lnxdx.
[2011, 12M]
10) Evaluate I=∬s(xdydz+dzdx+xz2)dxdy, where S is the outer side of the part of the sphere x2+y2+z2=1 in the first octant.
[2009, 20M]
11) Evaluate the double integral ∬ayxdxdyx2+y2 by changing the order of integration.
[2008, 20M]
12) Change the order of integration in ∫∞0∫∞xe−yydydx and hence evaluate it.
[2006, 15M]
13) Find the x-coordinate of the centre of gravity of the solid lying inside the cylinder x2+y2=2ax, between the plane z=0 and the paraboloid x2+y2=az.
[2005, 15M]
14) Find the centre of gravity of the region bounded by the curve (xa)2/3+(xb)2/3=1 and both axes in the first quadrant, the density being ρ=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>x2+y2. Compute ∭D2xdxdydz.
[2010, 20M]
2) Evaluate ∭(x+y+z+1)2dxdydz over the region defined by x≥0, y≥0, z≥0, x+y+z≤1.
[2001, 15M]