2

Theorem 3.4. Suppose each of f and g is integrable over the interval [a, b] then:for a number c cf is integrable over the interval [a, b]; furthermorefaaef = c f2. ...

Question

Theorem 3.4. Suppose each of f and g is integrable over the interval [a, b] then:for a number c cf is integrable over the interval [a, b]; furthermorefaaef = c f2. f + g is integrable over the interval [a,b]; furthermorefa f +9 = fa a,b] s+ f a.6]f . 9 is integrable over the interval [a,b]; furthermore if each of f' and is COIlinuous (henf() . 9(6) f(a)g(a) = Jan" f .9 + Jan f'4. If f(r) # 0,x € [a,b], $ is integrable over the interval [a, b]. If range(g) € domain( f) then f o

Theorem 3.4. Suppose each of f and g is integrable over the interval [a, b] then: for a number c cf is integrable over the interval [a, b]; furthermore faaef = c f 2. f + g is integrable over the interval [a,b]; furthermore fa f +9 = fa a,b] s+ f a.6] f . 9 is integrable over the interval [a,b]; furthermore if each of f' and is COIlinuous (hen f() . 9(6) f(a)g(a) = Jan" f .9 + Jan f' 4. If f(r) # 0,x € [a,b], $ is integrable over the interval [a, b]. If range(g) € domain( f) then f o g is integrable over an interval in the domain of g. Comments [re Theorem 3.4]: One of these is false (s0 it's not a theorem (yet) ) , add sufficient conditions for it to be true



Answers

Show that if $f(x)$ is integrable on every interval of real numbers, and if $a$ and $b$ are real numbers with $a<b,$ then a. $\int_{-\infty}^{a} f(x) d x$ and $\int_{a}^{\infty} f(x) d x$ both converge if and only if $\int_{-\infty}^{b} f(x) d x$ and $\int_{b}^{\infty} f(x) d x$ both converge. b. $\int_{-\infty}^{a} f(x) d x+\int_{a}^{\infty} f(x) d x=\int_{-\infty}^{b} f(x) d x+\int_{b}^{\infty} f(x) d x$ when the integrals involved converge.

All right, let's begin. Um, so we have a problem about improper integral roles with infinite limits. So I'm just going to remind you what this notation means. You know? What does it mean To integrate a function from some numbers, See, to infinity? Well, this is just the limit, uh, of of the integral of f over bigger and bigger intervals. So, um, this limit may may or may not exist When it does exist. We say that this improper integral converges eso dot Clarifies what? The notion of an integral to infinity conversion in the problem. And similarly, we can define, um, the integral for minus infinity to see as the limit. Hello of it. Intervals that increased towards minus infinity. So here we integrate for minus and to see, and we take them to infinity. So for part, they were asked to prove, and if and only if so first, let's suppose, um, that these two in a girls converge, meaning the corresponding limits actually exist so that these in it so that these integral is actually makes sense. Um, and what we want to prove that if we pick a different number, be, um, that these in a girl's also converge. So in the problem we assume, is less than B. So this only proves one implication will do the the second implication after this one. Um, so looking at the definition of improper integral, uh, what what we should do is shoes. Yeah, I am big enough. Yeah, So that b is less than M and minus, um is less than a and what we'll do eventually is take this big M to infinity. Okay, so, by a standard property vinegar rules the integral from, uh, be to I am fdx can be thought of was integrating from a, um fdx minus this intermediate integral. Right. So it is less than B. So, uh, if we integrate from 8 a.m. we've integrated, uh, over All right. We've integrated from a to B and then from B to em. So subtracting gives us this equation. And similarly, if we integrate for minus, um, Thio be well, this is like integrating from minus and to a followed by integrating from eight a b. Okay. Now, by assumption, uh, the limit of these two things as I'm approaches infinity actually exists so we can use the standard limit rules to conclude that. Yeah, the limit is I'm approaches Infinity of the integral from Vita mmf is the limit as I'm approaches infinity of in em, I think. And here, uh, this this part is independent to them, so taking the limit does nothing. So this doesn't change. And similar if the limit as, um approaches infinity of minus, um to be enough is the women, cause I'm approaches infinity 12. Well, let's see of this integral to a and again, the the A to be part doesn't depend on them. So these two things air true and well, uh, we get from the first equation bt Infinity of f yes, equals this, and we assume that this is well defined. So we get an expression for this integral from b and similarly from from the second equation. Yeah. Okay. This Okay? Um, yeah. So this proves that these two integral is actually do converge and they converge to these, um, these values. Okay, so this proves Well, I'm sorry that, um if these two in a girl's converge than the the implication is that these in the rules converge. Yeah. Now, if you look at the above our argument. It's pretty easy to see how toe go the other direction. So if you assume with that these two converge than by using basically the same trick, you can show that the integral from a to infinity and young girl from minus infinity to a also converge of these in a girls from being to be converged. So, um, maybe maybe here to clarify reverse implications. Similar? Yeah. Now for beef, uh, to show the required equality, all we do is just say, Well, if we integrate from minus, um t a than from 8 a.m. This is the same thing as integrating from well minus anto m. Whoops. Yes. And we can split that up. Aziz integrating from minus M to be followed by integrating from B J now taking limits. Well, we get exactly three equality in the question speaking. Yeah. Okay. Yeah. And this concludes, uh, the solution to this problem. Thank you for watching


Similar Solved Questions

5 answers
C) Give the numbers of protons and neutrons that make up the nucleus ofa 35Cl atom_
c) Give the numbers of protons and neutrons that make up the nucleus ofa 35Cl atom_...
5 answers
Consider a bipartite graph with a set L of nodes in the left column and set of nodes R on the right column, wherel L| RI: Prove or disprove the following claims: 11.62 The sum of the degrees of the nodes in L must equal the sum of the degrees of the nodes in R 11.63 The sum of the degrees of the nodes in L must be even. 11.64 The sum of the degrees of all nodes (that is, all nodes in LUR) must be an even number:
Consider a bipartite graph with a set L of nodes in the left column and set of nodes R on the right column, wherel L| RI: Prove or disprove the following claims: 11.62 The sum of the degrees of the nodes in L must equal the sum of the degrees of the nodes in R 11.63 The sum of the degrees of the nod...
5 answers
Calculate dyldt using the given information. 19) xy x= 12; dxldt =-3,x=2,y = A) -3 B) -920) x4/3+ y4/3 = 2; dv/dt=6,x=l,y= | A) 6 B) 6D) -6
Calculate dyldt using the given information. 19) xy x= 12; dxldt =-3,x=2,y = A) -3 B) -9 20) x4/3+ y4/3 = 2; dv/dt=6,x=l,y= | A) 6 B) 6 D) -6...
5 answers
(11) The principle argument of a complex number_ denoted by Arg(z) is the argument of z which is in-between and is inclusive, T is exclusive) Thus Arg(z <t. For example arg(~1+i) 3 +2kT, and Arg(- 1+i) 3; Arg(1-i) 1 What is the principle argument of ~1 = V3i? Find all In(-1 V3i) and the principle logarithm Ln(-l V3i) (use the principle argument)
(11) The principle argument of a complex number_ denoted by Arg(z) is the argument of z which is in-between and is inclusive, T is exclusive) Thus Arg(z <t. For example arg(~1+i) 3 +2kT, and Arg(- 1+i) 3; Arg(1-i) 1 What is the principle argument of ~1 = V3i? Find all In(-1 V3i) and the principl...
5 answers
32Prove the reduction formulacos"-1 , x sin € + n_1 cos"-21 drcos" € dxUse parU(a) to evaluate cos" € dx.Use padts a) and (b) to evaluate cos T dr
32 Prove the reduction formula cos"-1 , x sin € + n_1 cos"-21 dr cos" € dx Use parU(a) to evaluate cos" € dx. Use padts a) and (b) to evaluate cos T dr...
5 answers
(Total 2Opts) Assume that eR is a lower bound for set A 5 R Then; t = inf{A} if and only if, for Question every n € N there exists an element a € A such that a - 44t
(Total 2Opts) Assume that eR is a lower bound for set A 5 R Then; t = inf{A} if and only if, for Question every n € N there exists an element a € A such that a - 44t...
5 answers
11] 1-chlora-1-ethanoCNamc the following and state Lhe numbcr of hydrogens in cach Icalecule:Reactions:
11] 1-chlora-1-ethanoC Namc the following and state Lhe numbcr of hydrogens in cach Icalecule: Reactions:...
5 answers
Let $R$ be the region bounded by $y=x^{2}, x=1,$ and $y=0 .$ Use the shell method to find the volume of the solid generated when $R$ is revolved about the following lines.$$y=2$$
Let $R$ be the region bounded by $y=x^{2}, x=1,$ and $y=0 .$ Use the shell method to find the volume of the solid generated when $R$ is revolved about the following lines. $$y=2$$...
5 answers
Suppose that f(z) is differentiable for all & € (a,6) and that f-1(z) exists on (a,6). Call the inverse g(z). (Recall: This means that f(g(z)) = g(f (w)) = I, Prove that 9 (f(x)) (a < € < b). F() Let f (z) = cos + 2x Find 9 (1).
Suppose that f(z) is differentiable for all & € (a,6) and that f-1(z) exists on (a,6). Call the inverse g(z). (Recall: This means that f(g(z)) = g(f (w)) = I, Prove that 9 (f(x)) (a < € < b). F() Let f (z) = cos + 2x Find 9 (1)....
5 answers
4) Bottle #2 (mouth up with lid) pops when the lid is removedand the flame is above the bottle.What does this behavior tell you about the density of hydrogengas compared to air?
4) Bottle #2 (mouth up with lid) pops when the lid is removed and the flame is above the bottle. What does this behavior tell you about the density of hydrogen gas compared to air?...
5 answers
Describe 3 evolutionary trends that seem to have taken place in the evolution of the animals using examples.
Describe 3 evolutionary trends that seem to have taken place in the evolution of the animals using examples....
4 answers
A government “think tank” estimates that the typical teenagersends 50 text messages per day. To verify this statement, you calla sample of teenagers and ask them how many text messages they sentthe previous day. Their responses were as follows:5117547494454145203215942100At the 0.1 significance level, can you conclude that the meannumber is greater than 50? [10 marks]
A government “think tank” estimates that the typical teenager sends 50 text messages per day. To verify this statement, you call a sample of teenagers and ask them how many text messages they sent the previous day. Their responses were as follows: 51 175 47 49 44 54 145 203 21 59 42 100 ...
5 answers
A 0.820-kg ball is dropped from rest at a point 4.60 m above thefloor. The ball rebounds straight upward to a height of 1.50 m.Taking the negative direction to be downward, what is the impulseof the net force applied to the ball during the collision with thefloor?
A 0.820-kg ball is dropped from rest at a point 4.60 m above the floor. The ball rebounds straight upward to a height of 1.50 m. Taking the negative direction to be downward, what is the impulse of the net force applied to the ball during the collision with the floor?...
5 answers
Find the expected value for the random variable x having the probability function shown in the graph(10l351 10134 026 1oi246 (- Iolos Iod501Complete the probability distribution table below (Simplify your answers Type intege or decimals )X10 20 30 40 50P(x)
Find the expected value for the random variable x having the probability function shown in the graph ( 10l351 10134 026 1oi24 6 (- Iolos Iod 501 Complete the probability distribution table below (Simplify your answers Type intege or decimals ) X 10 20 30 40 50 P(x)...
5 answers
From the information given below, calculate the enthalpy of formation of one mole of CZHSOH liquid, by the reaction H2 (g) + € (s) 02 (g) CZHSOH() Given: CZHSOH() 02 (g) co2(g) +H2O (I) AH = -875.0kJ C(s)+ 02 (g) Co2(g) AH 3394.51kJ H2(g) + 02 (g) HZO () AH 2285.8 kJ(b) Calculate the quantun numbers for 7" , 12" , 18" and 25u electrons of manganese atomn (Mn.s)
From the information given below, calculate the enthalpy of formation of one mole of CZHSOH liquid, by the reaction H2 (g) + € (s) 02 (g) CZHSOH() Given: CZHSOH() 02 (g) co2(g) +H2O (I) AH = -875.0kJ C(s)+ 02 (g) Co2(g) AH 3394.51kJ H2(g) + 02 (g) HZO () AH 2285.8 kJ (b) Calculate the quantun ...
5 answers
Point charge Q1-30/Cis located at (x = by Q1and QZat theorigin 20my-0.z-O}and another point charge Q2--304C I-O.y-0.z-Ol is IS locared " 6-2nr20m 2-01- Tic43ONIC-336NIC+JONC33ONC
point charge Q1-30/Cis located at (x = by Q1and QZat theorigin 20my-0.z-O}and another point charge Q2--304C I-O.y-0.z-Ol is IS locared " 6-2nr20m 2-01- Tic 43ONIC -336NIC +JONC 33ONC...
5 answers
Use the Euclidean Algorithm to find d = gcd(1001,525) and use the extended algorithm to write d as an integer combina- tion d 10018 + 525t. Us- ing this immediately solve the equation IOOIx = in Z/525.
Use the Euclidean Algorithm to find d = gcd(1001,525) and use the extended algorithm to write d as an integer combina- tion d 10018 + 525t. Us- ing this immediately solve the equation IOOIx = in Z/525....

-- 0.070895--