Find $\int^{2}_{0} (x^2+1) d{\lfloor x \rfloor}$

Question

Find $$\int^{2}_{0} (x^2+1) d{\lfloor x \rfloor}$$ where $\lfloor x \rfloor$ denotes the greatest integer smaller than or equal to x.

I found this question in a book. After reading this question, I am now confused about what an integral is. Please try to clarify what this particular integral means and then solve it

It does look weird to me but I simply can't point out where it's wrong — Arpit Saxena, Dec 30 '17 at 01:47
@KennyLau It's a Stieltjes integral. Surely the floor function on $(a, b)$ is of bounded variation. — Sean Roberson, Dec 30 '17 at 02:03
@KennyLau Because it is not. Simply apply integration by part. — , Dec 30 '17 at 02:07
Thanks for pointing it out that it's a Stieltjes integral. I looked it up on Wikipedia and understood how to do the question. @SeanRoberson — Arpit Saxena, Dec 30 '17 at 02:10
@hellotinfish How can it be solved using integration by parts? Please add an answer — Arpit Saxena, Dec 30 '17 at 02:11
Sorry but I'm using my phone right now. I hate typing LaTeX on a phone. Maybe I'll do it later when I have my computer in front of me. — , Dec 30 '17 at 02:15
From what book did this come? What topic does that section cover? — gen-ℤ ready to perish, Dec 30 '17 at 03:42
@ChaseRyanTaylor I found it in a book which had questions from many topics. A question bank — Arpit Saxena, Dec 31 '17 at 03:17

score 5 · Accepted Answer · answered Dec 30 '17 at 01:47

5

Choose any partition $0=x_0<x_1<x_2<\cdots<x_n=2$ of the interval $[0,2]$ and note that the upper Reimann sum for this integral is $$\sum_{k=1}^n(x_k^2+1)(\lfloor x_k\rfloor-\lfloor x_{k-1}\rfloor)$$ Now note that if $x_k$ and $x_{k-1}$ both belong to either $[0,1)$ or $[1,2)$ then $\lfloor x_k\rfloor-\lfloor x_{k-1}\rfloor=0$. Can you do the rest?

answered Dec 30 '17 at 01:47

QED

12,644

I'm not really familiar with Reimann sums. Could you please elaborate further? – Arpit Saxena Dec 30 '17 at 01:49
1

@ArpitSaxena: Which definition of integrals are you working with, then? – hmakholm left over Monica Dec 30 '17 at 01:56
I'm just a beginner in integration. I'm not familiar with the different definitions of integrals. I was introduced to integration as an antiderivative and evaluating definite integrals by using the fundamental theorem of calculus. – Arpit Saxena Dec 30 '17 at 01:59
1

I got it after reading up on Riemann-Stieltjes integral. https://en.m.wikipedia.org/wiki/Riemann%E2%80%93Stieltjes_integral?wprov=sfla1 – Arpit Saxena Dec 30 '17 at 02:09

score 1 · Answer 2 · answered Dec 30 '17 at 05:31

Integration by parts: $$\int^2_0(x^2+1)d\lfloor x\rfloor=[(x^2+1)\lfloor x\rfloor]^2_0-\int^2_0\lfloor x\rfloor d(x^2+1)$$$$=10-\int^2_0\lfloor x\rfloor d(x^2+1)=10-2\int^2_0x\lfloor x\rfloor dx$$$$=10-2\int^1_0x\lfloor x\rfloor dx-2\int^2_1x\lfloor x\rfloor dx$$$$=10-2(0)-2\int^2_1x(1) dx=7$$

since $\lfloor x\rfloor=0$ for $x\in[0,1)$ and $\lfloor x\rfloor=1$ for $x\in[1,2)$

Find $\int^{2}_{0} (x^2+1) d{\lfloor x \rfloor}$

2 Answers2