# integration

2. ### How to integrate from a Fourier sine coefficient?

I try to solve a Fourier series (it is the solution of heat equation) as $$-100x = \sum_{n=1}^{\infty}b_n\sin n\pi x$$ $$b_n=2 \int_0^1{(-100x)\sin (n \pi x)}dx = -200 \int_0^1 x \sin (n \pi x) dx =(-1)^n \frac{200}{n \pi}$$ I know it is simple and stupid, but I do not understand how the...
3. ### how to change the order of integration? also steps of solving it?

You can find the question in the attachments, that's because I don't know how to type the equation in a perfect way by using the keyboard.
4. ### Continuous Random Variable Proof

Question: If X is a random variable for which P(X \geq 0)=0 and |\mu_x|<\infty, show that P(X<\mu t) \geq 1 - \frac{1}{t} for every t \geq 1. My TA told me to use the following information: \mu=\int_{-\infty}^{\infty}(x(f(x))dx Since P(X \geq 0)=0: \mu=\int_{0}^{\infty}(x(f(x))dx...
5. ### Volume of Hexagonal Pyramid - Using Integration from (0-h)

The Question: By using Riemann’s sum, synthesise a mathematical model for finding the exact volume of any ‘tepee’ tent of side s and height h. Here's what I have so far However, I am currently unsure if I am heading in the right direction, and am stuck at what comes next. Any help would be...
6. ### Riemanns Sum Problem

The Question: What I have so far: I have also proven that the vertical cross-sections result in the same formulae for Volume: The Questions: State one assumption that must be made and its associated effect in relation to finding a formula for the lightweight ‘pop-up’ tent. If the safety...
7. ### Mechanics problem: ratios of velocities, times and distances

I am working through an old mechanics textbook for UK A-level (1970s/80s) and am having difficulty with following problem, which is from an old A-level paper. I have managed to solve the first part of the question, but am floundering on the second and third parts. Help please! This is the first...
8. ### Why is that we use the boundaries from -1 to 1 instead of 0 to 1?

The given curve is rotated about the y-axis. Find the area of the resulting surface. $\sqrt[3]{x^2}+\sqrt[3]{y^2}=1 , 0\leq y \leq1$ We set up the surface area integral: \int_{-1}^{1} 2\pi x ds I understand how to find the arc-length(ds) and set up the integral but why...
9. ### Integral of 1/(1-x)

Yeah so basically we all know that the integral of 1/(1-x) is -ln(1-x)+c, but if I factor out -1 from the denominator before integration I seem to get -ln(x-1)+c. Can someone explain where the mistake is?
10. ### Volume of sphere with two cylindrical holes

How is one supposed to find the volume of a sphere with two cylindrical holes drilled into it, the two cylinders passing through the center and intersecting each other as well. Similar to this image. The problem in particular that I need help with has a sphere of radius 5 and cylindrical holes...
11. ### Help finding volume generated by revolving 2/(x+1)(2-x) around y axis

Hi. I need to find exactly the volume generated when the area bounded by y= 2/((x+1)(2-x)), the x and y axis, and x = 1 is revolved around the y axis. The correct answer should be 4pi*ln(2)/3 units. Thanks!
12. ### Hard Integral

How to calculate \int _{0}^{\infty} \frac {x \tan x \cos (\tan^2 x)}{x^2+1}dx
13. ### Help with leaving integratio in the form a+bsqrt3 and simultaneous equation please!!!

1.Find the values of p and q which satisfy the following simultaneous equations ∫from2to1(px^2 + q) dx = 9 ∫from2to1 (qx + p) dx = 6 2a) Expand (4sqrtx + 3)^2 DONE 16x + 24sqrtx + 9. b) Hence evaluate ∫from3to0 (4sqrtx + 3)^2 dx Giving your answer in the form a + bsqrt3 where a and b are...
14. ### integration

can some one help me with question 15. i am integrating it and trying this 1/3 (9)^3 -(1/3 (1)^3) and this gives me a answer of 242.66667 but the answer at the back of the book is 17 2/3
15. ### integration of trig function

Integrate sin(wt)sin(wt+s) Answer: (1/2)tcos(s) - (1/4w)sin(2wt+s)+c I would appreciate even a tip of how to start it correctly. I tried a lot to do this, but I can't find the correct answer.
16. ### integration

Find the area bound by the curve y=x^2-16 , the x-axis and the lines x=2 and x=5. i am trying to use the definite integral way ∫_2^5. but i am not getting the right answer. the right answer is 53/3
17. ### integration to find distance travelled

i am stuck on B(ii) how do you do this?
18. ### Complex linear integration

Hi everyone i need some help on some complex integration. I am struggling on how to go about these two questions as its a non-simple integral and we haven't covered this properly in Uni yet. On Question 3 i think I just have to present two options on whether z = w is included in C or not but...
19. ### Integration

\displaystyle \int e^{x^2}(x+x^3+2x^5)e^{x^4}dx
20. ### Integration

Hello, I have \int_0^1 \frac{1}{x} \log\left(\frac{1+x}{1-x}\right)=2\left(\frac{1}{1^2}+\frac{1}{3^2}+\frac{1}{5^2}+.....\right)=\frac{\pi^2}{4} i.e. \left(\frac{1}{1^2}+\frac{1}{3^2}+\frac{1}{5^2}+.....\right)=\frac{\pi^2}{8} I want to know how...