# What is Calculus 2: Definition and 117 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. ### Identifying variables from Riemann sum limits

Hi! I understand that this is an expanded Riemann sum but I'm having trouble determining its original form. I don't actually have any ideas as to how to find it, but I know that once I determine the original form of the Riemann sum, I will be able to figure out the values for a, b, and f. If...
2. ### Courses Should I take Calculus 2 my freshman year of college?

I self-studied Calculus 1 (AP Calc AB) in my sophomore year of high school and I got a 4 on the AP exam. I took the class Calculus 2 (AP Calc BC) in my junior year of high school. I did not take the exam at the end of the year. I didn't take any math in my senior year of high school. Because...
3. ### Troubles with an antiderivative

please tell me if i am right or wrong
4. ### Help explaining the chain rule please

I had already calculated the first partial derivative to equal the following: $$\frac{\partial y}{\partial t} = \frac{\partial v}{\partial x} \frac{\partial x}{\partial t} + \frac{\partial v}{\partial t}$$ Now the second partial derivative I can use the chain rule to do and get to...
5. ### Trapezoidal rule to estimate arc length error

I got the first part of it down, $$L=\int_1^5 \sqrt{1+(\frac{1}{x^2})}dx$$ I just want to know if it's right to make your ##f(x)=\sqrt{1+\frac{1}{x^2}}## then compute it's second derivative and find it's max value, for the trapezoidal error formula.
6. ### I Why can an infinite area have a finite volume or SA?

I have a calculus 2 midterm coming up and given the exam review questions, this seems like this question can potentially be on it. I've tried to look it up, but I always find the famous painters example, which I don't find satisfying.
7. ### I How to apply the disk/washer and shell methods

In Calculus II, we're learning about solids of revolutions and computing their volumes. I'm unsure when to apply the appropriate methods and how to make the correct partitions. Please tell me if my reasoning is correct: The disk/washer method is applied when your partitions are perpendicular...
8. ### Area of a bounded region using integration

In Calculus II, we're currently learning how to find the area of a bounded region using integration. My professor wants us to solve a problem where we're given a graph of two arbitrary functions, f(x) and g(x) and their intersection points, labeled (a,b) and (c,d) with nothing else given. I...
9. ### Does the series 4/k(lnk)^2 converge or diverge?

Homework Statement ##\sum_{k=0}^\infty \frac 4 k(\ln k)^2 ## Homework EquationsThe Attempt at a Solution I tried to solve it using the integral test but since it's not continuous it doesn't work.
10. ### MHB Calculus 2 (Power Series) when the limit is zero by root test

Hi guys! Here's a problem i was working on. I solved it by root test and got absolute value of x on the outside of the limit and the limit equaled zero. Is it wrong to multiply the outside absolute value by the zero I got from the limit? or is that okay? In general, when we are solving power...
11. ### Calculus 2 for Engineers: Riemann sums

Homework Statement a. Write down a Riemann sum for the integral ∫x3dx from 0 to 1. b. Given the following identity 13+23+33...+N3=(N(N+1)/2)2, show that the Riemann sums for ∫x3dx from 0 to 1 converge to 1/4. The Attempt at a Solution I believe I have gotten part a. I got ∑i^3/N^4 from i=0 to...
12. ### Finding area enclosed by the polar curve

Homework Statement Question attached in attachments Homework Equations Area enclosed by polar graph is ∫0.5r^2 where r is the radius as a function of angle theta The Attempt at a Solution I attempted to use the formula above and I subtracted the area of the inside from the outside but it...
13. ### Finding a the value of 30th derivative given power series.

Homework Statement The problem is attached as pic Homework Equations ∑(ƒ^(n)(a)(x-a)^n)n! (This is the taylor series formula about point x = 3)The Attempt at a Solution So I realized that we should be looking at either the 30th,31st term of the series to determine the coefficient. After we...
14. ### Calc 2 Integration Area Problem

Homework Statement Please help me solve the calc problem pictured! Homework Equations y=3-x^2 and y=x+1 The Attempt at a Solution My attempt is in one of the photos!
15. ### Finding the limits of integration and quadratic formula

Homework Statement Please look at the photo! Homework Equations -x^2+4x=x^2-6x+5 The Attempt at a Solution I got 2x^2-10x+5 but it says it's wrong
16. ### Calculus 2 - Center of Mass and Pappus Centroid Theorem

Homework Statement determine the center of mass of a thin plate of density 12 and whose shape is the triangle of vertices (1,0), (0,0), (1,1). Then, using the appropriate pappus theorem, calculate the volume of the solid obtained by rotating this region around the line x = -2. Homework...
17. ### Calculus 2 - Trig Integrals Question (Integrating cos^2x)

1. Here's the problem on trig integrating that I'm struggling with (Calculus 2 btw) 2. Wanted to see if I did everything right so far and what to do after all this. The part where I'm stuck is how to integrate (integral)cos^(2)udu and (integral)cos^(2)usin^(2)udu. I'm sure these are easy...
18. ### Triple integral using cylindrical coordinates

Homework Statement The first part of the question was to describe E the region within the sphere ##x^2 + y^2 + z^2 = 16## and above the paraboloid ##z=\frac{1}{6} (x^2+y^2)## using the three different coordinate systems. For cartesian, I found ##4* \int_{0}^{\sqrt{12}} \int_{0}^{12-x^2}...
19. ### Mathematically rigorous Calculus 2 book

So, i am currently studying physics in a brazilian university. I am going to have a Calculus 2 course which, in Brazil, covers Ordinary Differential Equations and multi-variable differential calculus. So which challenging and rigourous books would you guys recommend for that? Thanks for the...
20. ### Is it possible to self study calculus 2?

I am a high school student entering 11th grade soon. I will be taking calculus ab (calc 1) in school, but I am wondering if it is possible to self study calculus 2 and take it over the summer at a cc?(and do well!) If everything goes right I want to do calculus 2 the summer after junior year and...
21. ### MHB How do you solve this calculus 2 integration problem?

I am currently studying calculus 3. However, once in a while, I like to review single variable calculus. I decided to tackle the following calculus 2 integration problem: ∫ tan^3 x dx Solution: Separate into two tangent functions. (tan^2 x)(tan x) Rewrite tan^2 x as sec^2 x - 1 using its...
22. ### Need help finding the limit of a function

Homework Statement Calculate limit as x approaches infinity of (e^x - x^3) Homework Equations ln e^x = x e^(ln x) = x The Attempt at a Solution I tried substituting x = ln e^x and got (e^x - (ln e^x)^3). I'm pretty much lost and this is my only attempt so far. I'm thinking that this is an...
23. ### Need help evaluating an improper integral as a power series.

Homework Statement Evaluate the indefinite integral as a power series. What is the radius of convergence (R)? ##\int x^2ln(1+x) \, dx## Book's answer: ##\int x^2ln(1+x) dx = C + \sum_{n=1}^\infty (-1)^n \frac {x^{n+3}} {n(n+3)}; R = 1## Homework Equations Geometric series ##\frac {1} {1-x} =...
24. ### Show Wolfram Alpha's answer is equivalent to my answer.

Homework Statement Integrate x2(2+x3)4dx. Show that Wolfram Alpha's answer is equivalent to your answer. Homework Equations No equations besides knowing that the integral of xpower is 1/power+1 * xpower + 1 The Attempt at a Solution So I have the answer to the integral by hand as (2+x3)5)/15...
25. ### Work problem -- lifting water out of tanks

< Mentor Note -- thread moved to HH from the technical math forums, so no HH Template is shown > I've encountered 2 problems in a row that involve lifting water out of tanks and finding the work needed. I am getting the incorrect answer. w = ⌠ab pgA(y)D(y)dy here is one of the problems: A...
26. ### I Calculating Work Needed to Stretch a Spring: 100J to 0.75m

"It takes 100J of work to stretch a spring 0.5m from its equilibrium position. How much work is needed to stretch it an additional 0.75m." Attempt: w = ⌠abF(x)dx work = F x D 100J = F x 0.5m F = 200J 0.75 + 0.5 = 1.25 w = ⌠0.51.25 200dx w = 150 J The correct answer: w = 525 J what did I do...
27. ### Fluid Pressure & Force Calculus 2

Homework Statement A marine biology observation pod is being designed. It will be submerged, with vertical windows of various sizes and shapes. Calculate the total force being exerted on each of the windows described below. Density of water is 1000 kg/m^3...
28. ### How Can I Simplify This Integral Using Green's Theorem?

Homework Statement I have a linear integral (e^xsiny-2)dx + (e^xcosy+x^2)dy y≥0 2x=x^2+y^2 I used Green's theorem and got: ∬ (e^xcosy+2x) - (e^xcosy) dy dx x bounds: from 0 to 2 y bounds: from 0 to sqrt(2x-x^2) After solving all that stuff I get to: ∫ (2x) (sqrt(2x-x^2)) dx x bounds: 0 to...
29. ### For the life of me I can't solve this integral sqrt(1-x^3)

Homework Statement Hi, I have a double integral with square root of 1-x^3 that I can't solve. Here is the integral and attempt at the solution. I get stuck! Homework Equations I am in Calc II. Mostly we used integration by parts, or substituting t=something for most "difficult" integrals...
30. ### I Find the formula to express the infinite series....

The problem is to find the general term ##a_n## (not the partial sum) of the infinite series with a starting point n=1 $$a_n = \frac {8} {1^2 + 1} + \frac {1} {2^2 + 1} + \frac {8} {3^2 + 1} + \frac {1} {4^2 + 1} + \text {...}$$ The denominator is easy, just ##n^2 + 1## but I can't think of...
31. ### I Lebesgue measure and Fourier theory

Hi everyone, in this days i was seeing a little of Fourier series and transform, and i wondered if it was necessary to better understand before the measure and Lebesgue integral before studying it. Or it's not necessary?
32. ### Courses Calculus 2 Online Course Required

Hey everyone! I started back at my university after a 3 & 1/2 year break and changed my major from Journalism to Physics last May. I am coming up on my finals in a couple of weeks for the Spring Semester and then I will be taking a class over the summer and doing research. My grades are great...
33. ### Find Taylor Series from a function and its interval of convergence

Let f(x) = (1+x)-4 Find the Taylor Series of f centered at x=1 and its interval of convergence. \sum_{n=0}^\infty f^n(c)\frac{(x-c)^n}{n!} is general Taylor series form My attempt I found the first 4 derivatives of f(x) and their values at fn(1). Yet from here I do not know how to find the...
34. ### Sequence Convergence/Divergence Question

Homework Statement Determine which of the sequences converge or diverge. Find the limit of the convergent sequences. 1) {asubn}= [((n^2) + (-1)^n)] / [(4n^2)] Homework Equations [/B] a1=first term, a2=second term...an= nth term The Attempt at a Solution a) So I found the first couple of...
35. ### 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...
36. ### Which Book is Best for Studying Analysis 2 and Multivariable Calculus?

Hi at all, what is the better book for study analysis 2, if possible also with exercises...
37. ### Not enjoying Calculus 2 at all, advice?

Hey everyone, First of all, let me start out by saying that I am not enjoying this class, not because I'm not doing well, but frankly because I find it boring. It's not that I'm not challenged, I am. The material can be meticulous. But, it just seems like a lot of algebraic manipulations and...
38. ### Tank Mass Balance Calculation for Designing w/ Automatic Float Switch

I'll be using a mass balance calc for a tank design at work. We are designing a tank that has an automatic float switch on it, attached to a valve which will start filling the tank when the water level drops below a certain point. This tank is intended to drain quickly (flow rate not known at...
39. ### AP Calc BC Prep in 2 Months: Is It Possible?

Hello I am currently a sophomore in high school and am looking to graduate early next year. I need to take 4 additional classes this summer at my local community college in order to satisfy all the courses I need in order to graduate early. To give my self a boost I was looking into taking the...
40. ### Calculus 2 math project: Calculating orbital work

1. My calculus 2 math teacher has us doing a project, involving calculus, on any topic we choose. I chose the topic of calculating work, as in Force x Distance. Specifically calculating work done by lifting a mass into orbit, or to escape velocity. A method is used in my textbook for calculating...
41. ### 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...
42. ### Does Calculus 2, Series Stuff Help w/ CS?

I am in calc2 now and I just can't get excited about this series stuff. We went over many methods of testing for convergence/divergence and finally moved on to polar coordinates. Is series important in any type of CS field besides, I would imagine, creating software to solve series problems?
43. ### Solving Shell Motion with Initial Velocity of 60 m/s

I've put the problem statement below and worked it out. I typically don't post questions like this as they're a lot to go through, but I am wondering if I have worked the problem correctly as my book does not have the solution and I feel like I am not understand the material correctly. 1...
44. ### Definite integral involving partial fractions

Homework Statement Homework Equations trigonometric identities The Attempt at a Solution I did a trig substitution of u=tan(θ/2) and from that I could substitute cos(θ) = 1-u2/1+u2 dθ = 2/(1+u2) du = 1/2 sec2(θ/2) dθ I simplified a bit and changed the bounds to get 2du/(5u2 + 1)(1 + u2)2...
45. ### Calculus 2 interval of convergence -- checking end points

Homework Statement...
46. ### Order of concepts taught in calculus 1 & 2

Hello everyone. I am about to start Calculus 1 and then Calculus 2. I want to get an idea of how the order of concepts of these to classes are generally laid out.Thanks