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 x0, x1, x2,…xn such that a=x0≤x1≤⋯≤xn−1≤xn=b and is denoted by P={x0,x1,x2,…,xn}.
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The upper Riemann sum is given by:
U(P,f)=n∑1MiΔxiwhere Mi=sup{f(x):xi−1≤x≤xi} and Δxi=xi−xi−1
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The lower Riemann sum is given by:
L(P,f)=n∑1miΔxiwhere mi=inf{f(x):xi−1≤x≤xi} and Δxi=xi−xi−1
Since f is bounded, ∃ real numbers m and M such that m≤f(x)≤M ∀ x∈[a,b]. Thus, for every partition P,
m(b−a)≤L(P,f)≤U(P,f)≤M(b−a)
We define:
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The upper Riemann integral of f over [a,b] as ∫bafdx=infU(P,f).
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The lower Riemann integral of f over [a,b] as ∫bafdx=supL(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 ∫bafdx.
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] ∀ x>a.
If limx→∞∫xaf(t)dt=L, for some L∈R, then we say that the improper integral of first kind, ∫∞af(t)dt, converges to L, and is denoted by ∫∞af(t)dt=L.
Otherwise, the improper integral ∫∞af(t)dt is said to diverge.
Example: The improper integral ∫∞11t2dt converges because ∫x11t2dt=1−1x→1 as x→∞. On the other hand, the improper integral ∫∞11tdt diverges because limx→∞∫x11tdt=limx→∞logx→∞ as x→∞.
We can show that ∫∞11tpdt converges to 1p−1 for p>1 and diverges for p≤1.
Comparison Test and Limit Comparison Test
Suppose f is integrable on [a,x] ∀ x>a. Then, we state following two theorems:
Comparison Test: Let 0≤f(t)≤g(t) ∀ t>a. Then,
If ∫∞ag(t)dt converges, →∫∞af(t)dt converges.
Limit Comparison Test (LCT): Let f(t), g(t)≥0 ∀ x>a.
If limt→∞f(t)g(t)=c, where c≠0, then both the integrals ∫∞af(t)dt and ∫∞ag(t)dt converge either converge, or diverge together.
If c=0, the convergence of ∫∞ag(t)dt implies the convergence of ∫∞af(t)dt.
Example: The integral ∫∞1sin1tdt diverges by LCT because sin1t1t→1 as t→∞. On the other hand, for p∈R, ∫∞1e−ttpdt converges by LCT because e−ttpt−2→0 as x→∞.
Fundamental Theorems Of Integral Calculus
First Fundamental Theorem of Calculus
Let f be integrable on [a,b]. For a≤x≤b, let F(x)=∫xaf(t)dt. Then, F is continuous on [a,b].
Also, if f is continuous at x0, then F is differentiable at x0 and F′(x0)=(x0).
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 ∫baf(x)dx=F(b)−F(a).
Riemann Sum
Let f:[a,b]→R and let P={x0,x1,…,xn} be a partition of [a,b].
Let ck∈[xk−1,xk], k=1, 2,.., n. Then, corresponding to the partition P and the intermediate points ck, a Riemann sum for f is defined as S(P,f)=∑nk=1f(ck)Δxk.
Also, the norm of P is defined as ‖P‖=max1≤i≤nΔxi.
PYQs
Riemann Integral
1) Evaluate
∫∞0tan−1(ax)x(1+x2)dxa>0,a≠1[2019, 10M]
2) Prove the inequality: π29<∫π2π6xsinxdx<2π29
[2018, 10M]
3) Is the function
f(x)={1n,1n+1<x≤1n0,x=0
Riemann integrable? If yes, obtain the value of ∫10f(x)dx
[2015, 15M]
4) Integrate ∫01f(x)dx, where f(x)={2xsin1x−cos1x,x∈[0,1]0,x=0
[2014, 15M]
5) Let f(x)={x22+4 if x≥0−x22+2 if x<0
Is f Riemann integrable in the interval [−1,2]? Does there exist a function g such that g′(x)=f(x)? Justify your answer.
[2013, 10M]
6) Let [x] denotes the integer part of the real number x, i.e., if n≤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 ∫2−1([x]2+3)dx.
[2013, 10M]
7) Give an example of a function f(x), that is not Riemann integrable but |f(x)| is Riemann integrable. Justify your answer.
[2012, 20M]
8) Show that the function f(x) defined as
f(x)=12n, 12n+1≤x≤12n, n=0,1,2,…… and f(0)=0 is integrable in [0,1], although it has an infinite number of points of discontinuity. Show that ∫10f(x)dx=23.
[2004, 12M]
9) A function f is defined in the interval (a,b) as follows:
f(x)={1q2 when x=pq1q3 when x=√pq
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=36−4x2−9y2.
[2007, 20M]
2) Find the volume of the ellipsoid
x2a2+y2b2+z2c2=1[2006, 20M]
3) Evaluate
∭ln(x+y+z)dxdydz[2005, 12M]
4) Find the volume bounded by the paraboloid x2+y2=az, the cylinder x2+y2=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 ρ depending on the distance R from the centre and is given by: ρ=k(2a−R), where k is a constant.
Find the mass of the hemisphere, by the method of multiple integrals.
[2002, 15M]
7) Evaluate ∭(ax2+by2+cz2)dxdydz taken throughout the region x2+y2+z2≤R2.
[2001, 15M]
Improper Integrals
1) Discuss the convergence of ∫21√xlnxdx.
[2019, 15M]
2) Test the convergence of the improper integral ∫∞1dxx2(1+e−x).
[2014, 10M]
3) Examine the convergence of ∫10dxx1/2(1−x)1/2.
[2006, 12M]
4) Show that ∫∞0e−ttn−1dt is an improper integral which converges for n>0.
[2005, 30M]
5) Show that ∫∞0dx1+x2sin2x is divergent.
[2003, 20M]
6) Prove that the integral ∫∞0xm−1e−xdx is convergent if and only if m>0.
[2002, 12M]
7) Show that ∫π/20xnsinmxdx 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 ∫10f2(x)dx=1. Prove that ∫10xf(x)f′(x)dx=−12.
[2012, 15M]
2) Let f be a continuous function on [0,1]. Using first Mean Value theorem on Integration, prove that limn→∞∫10nf(x)1+n2x2dx=π2f(0).
[2008, 15M]