What is Calculus ii: Definition and 107 Discussions

Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations.
It has two major branches, differential calculus and integral calculus; the former concerns instantaneous rates of change, and the slopes of curves, while integral calculus concerns accumulation of quantities, and areas under or between curves. These two branches are related to each other by the fundamental theorem of calculus, and they make use of the fundamental notions of convergence of infinite sequences and infinite series to a well-defined limit.Infinitesimal calculus was developed independently in the late 17th century by Isaac Newton and Gottfried Wilhelm Leibniz. Today, calculus has widespread uses in science, engineering, and economics.In mathematics education, calculus denotes courses of elementary mathematical analysis, which are mainly devoted to the study of functions and limits. The word calculus (plural calculi) is a Latin word, meaning originally "small pebble" (this meaning is kept in medicine – see Calculus (medicine)). Because such pebbles were used for counting (or measuring) a distance travelled by transportation devices in use in ancient Rome, the meaning of the word has evolved and today usually means a method of computation. It is therefore used for naming specific methods of calculation and related theories, such as propositional calculus, Ricci calculus, calculus of variations, lambda calculus, and process calculus.

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  1. Needassistance0987

    Comparison test to determine convergence

  2. S

    What type of mathematics did you find the most difficult?

    Mathematics tends to be more progressive than any other field. But I've heard some people say that some math classes that they took were more difficult than the most advanced math classes that they took. For instance, I've heard people say that Calculus II is more difficult than Calculus III...
  3. R

    Correct Parametrization for Calculating Area of a Tube?

    Hi, I'm trying to find the area of this tube using ##\int \int ||\vec{N}|| ds d\theta##. However, I get 0 as result which is wrong. So at this point, I'm wondering if I made a mistake during the parametrization of the tube. This is how I parametrized the tube. ##S(s, \theta) = (cos(s), sin(s)...
  4. D

    Calculus II: Convergence of Series with Positive Terms

    Homework Statement https://imgur.com/DUdOYjE The problem (#58) and its solution are posted above. Homework Equations I understand that I can approach this two different ways. The first way being the way shown in the solution, and the second way, which my professor suggested, being a Direct...
  5. J

    MHB Testing Convergence, Calculus II

  6. CrunchBerries

    Testing Calculus II Exam points question

    I wrote a Calc II midterm today. Things went quite well, except for one question. I left it blank, as I did not have sufficient knowledge to write anything that made any sense. The question was worth 10% of the exam mark. This question was about a part of a section we covered, but this part had...
  7. gsmtiger18

    Limits of sequences involving factorial statements

    Homework Statement I have to determine whether or not the following sequence is convergent, and if it is convergent, I have to find the limit. an = (-2)n / (n!) In solving this problem, I am not allowed to use any form or variation of the Ratio Test. 2. The attempt at a solution I was...
  8. visuality

    Calculus Calculus II book to really understand

    I'm taking Calculus II and my class is a flipped classroom so I am excited to finally be able to learn in a way that works for me. However, I am having some trouble finding resources to help me truly understand the material. Ideally, a book would introduce topics in a way that explains the...
  9. R

    Calc II Tips: Revise Calc 1 & Trig Before Semester Starts

    Hey there, I will be taking calculus 2 this spring semester and I would like to know what topics from calc 1 I should be reviewing, I did receive a B+. I am also aware that trig is highly involved and my trig skills are not that good, so if there's anything from trig that is mandatory to review...
  10. M

    Can I learn Calculus II on my own?

    I don't know how this happened by college has given me credit for Calc II even though I didn't take it. I took AP calc I but not AP calc II, but the college has given me credit for both. I am a Civil engineering major and I am wondering if I should try to audit the class next semester (meaning...
  11. TyroneTheDino

    Integrating Using a Substituation

    Homework Statement Use the substitution ##u=\frac{\pi} {2}-x## evaluate the integral ##\int_0^\frac {\pi}{2} \frac {\sin x}{\cos x + \sin x}dx##. Homework Equations [/B] ##\cos (\frac {\pi}{2}-x)=\sin x## The Attempt at a Solution [/B]I start by plugging "u" into the equation making the...
  12. K

    Calculus 2 Integrals Homework Solutions

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  13. Z

    Volume of a Solid Revolved About X-Axis

    I'm trying to practice for my final. The sample problem is: "Find the volume of the solid generated when the region bounded by y = x4and y = x1/3, 0<=x<=1, is revolved about the x-axis." To start, I set the two y equations equal to each to find the points of intersection. x4 = x1/3, : raise...
  14. M

    Calculus II Differential Equation Question

    Homework Statement We are just starting to learn about basic differential equations in Calc 2 and I learned about separable differential equations but I'm not familiar with this style, here's the question: Given the differential equation of the form ay"+by'+y=0, find constants a and b so that...
  15. Abner

    Representing functions as power series

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  16. L

    Calculating Arc Length of a Curve: A Calculus II Problem

    Homework Statement Find the exact length of the curve: y= 1/4 x2-1/2 ln(x) where 1<=x<=2 Homework Equations Using the Length formula (Leibniz) given in my book, L=Int[a,b] sqrt(1+(dy/dx)2) I found derivative of f to be (x2-1)/2x does that look correct? The Attempt at a Solution I found f'...
  17. L

    Calculus II Problem: Describe the region

    Homework Statement Sorry, this should say "Calculus III problem". Describe the region that contains all planes passing through (4, 5, 6) that are parallel to <1, 2, 3>. Homework Equations The Attempt at a Solution I cannot imagine how to answer this. I expect that there are multiple planes...
  18. S

    MHB Calculus III is SO MUCH EASIER than Calculus II

    I'm in Calculus III now, let me just tell you all - it is SO MUCH EASIER than Calculus II. I went through hell in Calculus II. Calculus III is so much easier that it's kind of ridiculous. What's Calculus 4 like?
  19. Y

    Tips for calculus II materials

    Good evening guys, next period I'm going to take Calculus II at the University, but turns out that in my university they split it in Calculus II-A and II-B. The topics covered in each are basically those: Calculus II-A: Definite Integral, Techniques of integration, Improper integrals, first...
  20. A

    MHB Calculus II Work and Density Question

    Hey guys, I'm not too sure of my answer for this question. Knowing the the area is πr^2 for a single slice, I setup an integral with radius =5 and got $ \int_{0.3}^{1.5} 245,000πx \,d $ ultimately leads to 264,000π Joules for the answer. Of course, this is bearing in mind that F=mg and...
  21. A

    MHB Understanding the Fundamental Theorem of Calculus in Calculus II

    Hey guys, I'm confused as to how these work. For instance, I'm assuming that the FTC must be used, where the formula used would be f(b)-f(a), correct? If so, I got 4x^3 radical(2-x^4) -2 for the first one. The second one is 3x^2 (sin(x^3)) + sincosx * cosx. Thanks in advance.
  22. A

    MHB Calculus II U-Substitution Questions

    This thread is only addressing question 6. 6a and 6b seemed relatively simple, with 6c seeming to be the most troublesome. I got 1/4 arcsin (x^2 / 2 ) + c for 6a after substituting u for x^2. As for 6b, I got (x^2)/2 -2ln(x^2 +4) + c after using algebraic division and taking u=x^2 + 4...
  23. A

    MHB Calculus II Integral Convergence and Divergence Questions

    For a,b, and c respectively, I got divergent (to -infinity), convergent (to π/6), and divergent (to infinity, since the first part's sum is 1/3, but lim negative infinity gives infinity, thus the summation of the two integrals gives a divergent integral). I'm sure these are right, but I'd...
  24. A

    MHB Calculus II Simpson's, Trapezoidal, and Midpoint Rule

    Hey guys, I'd appreciate some tips for this question. For question 7a, I used 1/2 * 4/8 [1f(x) + 2 + 2 + 2 + 1.. etc] and computed 1.732865 as the answer. For part b, I used 1/4, 3/4.. (etc.).. up to 15/4 for the midpoints, added the f(x) values of those points together, and divided...
  25. A

    MHB Calculus II Volumes of Revolution and Basic Integration Questions

    Hey guys, I have a couple of questions about the problem set I'm doing at the moment. Although I was able to solve most of these, I'm doubting quite a few of my responses. http://i.share.pho.to/f7d7efe6_o.pnghttp://i.share.pho.to/82c05629_o.png http://i.share.pho.to/d6f76bb6_o.png...
  26. A

    MHB Basic Calculus II Integral Questions - Riemann Sums, Absolute Integrals, etc.

    Hey guys, I'd appreciate some help for this problem set I'm working on currently The u-substitution for the first one is somewhat tricky. I ended up getting 1/40(u)^5/2 - 2 (u) ^3/2 +C, which I'm not too sure about. I took u from radical 3+2x^4. For the second question, I split the integral...
  27. S

    MHB Volumes by Cylindrical Shells (Calculus II)

    Quick question, may seem rather dumb - but I just want to make sure of something.. Question: Find the volume of the solid obtained by rotating about the y-axis the region between y = x and y = x^2 so when I am setting up my integral am I correct in saying TOP - BOTTOM i.e. --> \int^1_0 (2\pi...
  28. sheldonrocks97

    Intro to Linear Algebra vs Calculus II

    I am thinking of taking an Intro to Linear Algebra course at a community college to save money while I am at a university. It is a 200 level course just like Calc II and I just was wondering if it was harder than Calc II. I really struggled in that class and I am taking it again this semester so...
  29. sheldonrocks97

    I am Struggling in Calculus II, any Tips?

    I have a very hard teacher for my Calculus II class and I think I might not pass. Don't get me wrong, he's a great teacher, but he demands a lot of work to get a good grade. I currently have a 65.2% and to get a C we need a 64% (he grades on a curve). I feel like my head is just above water and...
  30. I

    Calculus II please explain integral test

    i understand the reason for if the evaluated integral converges then the series also converges. However, just because the evaluated integral diverges, why does this automatically mean that the related series also diverges? the integral consists of every number of the +x axis when evaluating...
  31. I

    Calculus II i don't understand the proof for the limit comparison test

    would someone please care to reword this proof for me? http://en.wikipedia.org/wiki/Limit_comparison_test it talks about ε, which is not even defined and then n0, which is again not defined, what the hell are all these variables... I'm sure someone here could do a better job organizing...
  32. D

    What are some tips for succeeding in Calculus II?

    Hello users! Im taking calculus 2 for summer(staring in 3 weeks), and it's the first time I take the course. Im studyng by my self, but I don't have a syllabus or something to follow. I already Covered this parts: Integration by Parts Trig Sub Trigonometric Integrals Improper...
  33. 1

    Calculus II- Pressure and Force Question

    Homework Statement Find the force acting on one end of a cylindrical drum with 3m. If the rim is submerged horizontally into water so that the bottom is 10m deep. Homework Equations A(x) - Area p(x) - Pressure = density * Depth g= 9.8 A(x) =2\sqrt{9-x^{2}} P(x) = 1000(7+x) * g...
  34. K

    Calculus I review for Calculus II

    I'm scheduled to take Calculus II this Fall semester, but it will have been over a year since I'd taken Calculus I. I am aware that I need to review, but what are some are the major concepts I should review to adequately prepare for the infamous Calc II?
  35. D

    Calculus II question, prove ∫ ≤ 14/3 without evaluating the integral

    Homework Statement Show that ∫(0,3) √(x+e^-x)dx ≤ 14/3 (hint: do not attempt to evaluate the integral)when looking at the integral from 0 to 3 on the graph I can see that this is true, but not sure how to go about showing this without evaluating the integral, any help as to how to go about...
  36. J

    Struggling in Calculus II? Need study tips for integration and substitution?

    Hello everyone. I recently started taking Calculus II this semester at my community college and I am having a hard time. I didn't do very well on the first exam and now we are getting into Integration by Parts, Trigonometric Substitution, and Partial Fractions. Does anyone have any tips on how...
  37. F

    Calculus II Proof by induction problem

    Homework Statement Let a1=2, a2=0, a3=-14 and an+1=9an-23an-1+15an-2 for n>= 3. Show by induction that an=3n-1-5n-1+2 for n>=1.Homework Equations The Attempt at a Solution Ok this is annoying to type... Base Case is true, which is the easy part. The induction part I am not going to list all...
  38. F

    Calculus II - Real numbers proofing

    Homework Statement Show that |a-b|<= |y-a|+|x-y|+|x-b|, for all x,y in ℝ Homework Equations The Attempt at a Solution |a-b| <= |y-a+x-y+x-b| (correct? Not sure about this one...is it not part of the triangle rule?) |a-b| <= |2x-b-1| |a-b+2x-2x| <= |2x-b-1| |a-2x| + |2x-b| <=...
  39. B

    Calculus II - Improper Integral Problem

    Homework Statement Evaluate the integral: ∫(0 to ∞) [dv/((1+v^2)(1+tan^-1(v))] Homework Equations U-substitution, taking limit to evaluate improper integralsThe Attempt at a Solution http://imgur.com/CjkRF As you can see in the image, I try u-substitution and then take the integral. I end...
  40. M

    So I've decided to deliberately fail Calculus II?

    Hi folks, long story short I failed my first Calculus II exam back in October, but since I couldn't get my money back, or get to drop the course, I've decided to stick with it at least to see the material and perhaps pass it on the final. However, I'm likely to fail the second midterm tomorrow...
  41. S

    Find an equation in standard form for the conic. Calculus II

    Homework Statement Find an equation in standard form for the conic , a)Ellipse: vertices (0,0) & (0,8), foci (0,2) & (0,6). b) hyperbola: vertices (-3,0) & (3,0), asymptotes y= ±2x Homework Equations Ellipse: (x-h)^2/b^2+(y-k)^2/a^2=1, vertices ± a from center, foci ± c from center...
  42. C

    Calculus II - Alternating Series Test - Convergent?

    Hello! I was working some practice problems for a Calc II quiz for Friday on the alternating series test for convergence or divergence of a series. I ran into a problem when I was working the following series, trying to determine whether it was convergent or divergent: Homework Statement ∞...
  43. A

    Calculus II - closed function question

    there is a function: F ( x, y, z) = 2ln (xz) + sin ( xyz) − y^2 = 0. the func is defined by the closed function z=f(x,y) and provides : f(1,0)=1 we define: g(t)=f(t,1-t^6) . where t is very close to 1. I have to find g'(1)Homework Equations I tried to to do like that: find F'x and F'z and did...
  44. U

    Advice on learning Calculus II on my own.

    Hey everyone, I'm currently enrolled in an AP Calculus AB class at my high school. However, for Cornell Engineering I need a 4 or 5 on the AP Calculus BC exam. I've started to teach myself Calc II, but does anyone have any advice on how to succeed in doing this? I am using Paul's online math...
  45. QuarkCharmer

    Difficult Calculus II integral

    Homework Statement \int_{-1}^{0} \frac{e^{\frac{1}{x}}}{x^{3}}dx Solve the integral and determine if it converges/diverges ect. Homework Equations The Attempt at a Solution \int_{-1}^{0} \frac{e^{\frac{1}{x}}}{x^{3}}dx \lim_{t \to 0}\int_{-1}^{t} \frac{e^{\frac{1}{x}}}{x^{3}}dx u =...
  46. G

    Calculus II - Determining if Infinite Sequences Converge

    Hi, I'm studying infinite series and was wondering if someone could recommend me a gigantic list of examples of series and proofs of weather they converge or not.
  47. G

    How do we determine the convergence at the endpoints for the series (x-1)^n/n^3?

    Hi, I don't really need help with a problem, just having some troubles understanding something. Find the interval of convergence of the series sigma[n=0,inf] (x-1)^n/n^3 by the root test I got that |x-1|<1 and that 0<x<2 I than have to plug in these values (0 and 2) to see if the...
  48. G

    Is this series convergent or divergent?

    Homework Statement Prove that sigma[n=0,inf] ((-1)^n n)/(n+1) diverges Homework Equations The Attempt at a Solution I'm unsure how to do this I tried applying the alternating sereis test but when I did so I got (n/(n+1))' = 1/(n+1)^2 so I can't say that the terms are...
  49. G

    Interval of Convergence for Infinite Series with Ratio Test

    Homework Statement Find the Interval of Convergence for the given series. Check the endpoint behavior carefully sigma[n=0,inf] (n (x-2)^n)/( (n+1)4^n ) Homework Equations The Attempt at a Solution I was following along with the answer key and they used the ratio test... The...
  50. G

    Calculus II - Calculus in Polar Coordinates

    Homework Statement Find the points at which the following polar curves have a horizontal and vertical tangent line. (a) r = 3 + 6 cos(theta) Homework Equations The Attempt at a Solution x = r cos(theta) = (3 + 6 cos(theta)))cos(theta) = 3cos(theta) + 6 cos(theta)^2 y =...
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