3
votes
4answers
75 views

Integration for finding the Arc Length of Circle $x^2+y^2=a^2$

Question: Find the arc length of the circle given by $x^2+y^2=a^2$. $Ans = 2\pi a$ How to obtain the ans? I have no ideas after doing the following thing. Thank you for your ...
2
votes
2answers
62 views

Find $\int_\Gamma\frac{2z+j}{z^3(z^2+1)}\mathrm{d}z$ where $Γ:|z-1-i| = 2$

pls, some ideas for integral solution (residue theory)? $$\int_\Gamma\dfrac{2z+j}{z^3(z^2+1)}\mathrm{d}z$$ Where $Γ:|z-1-i| = 2$ is positively oriented circle. Thx, for help!
1
vote
2answers
48 views

Circle Area formula question

Take a peek at the following proof Everything makes sense but one thing: how did they determine that $\sqrt{\cos^2\theta}$ was positive and not negative? Thanks.
0
votes
2answers
62 views

Paramtrizing a counterclockwise circle vs. a clockwise one

Does it make a different when you parametrize a counterclockwise full circle and a clockwise circle in the complex plane? For example, I am looking at computing an integral $\int_\gamma ...
2
votes
1answer
167 views

Numerical integration of a region bounded by an ellipse and a circle

Consider an ellipse (say with major axis $a$ and minor axis $b$) centered at origin with a concentric circle of radius $R$. Area of the region between the circle and the ellipse is $$A = \pi R^2 ...
4
votes
2answers
110 views

Evaluate $I = ∫∫ 1/((x^2 + y^2)^{n/2}) dxdy$

Evaluate the double integral $$ I = \int\int_D \frac{1}{(x^2 + y^2)^{n/2}} dxdy .$$ where $n$ is an integer and $D$ is the region of the plane bounded by two circles centered on the origin and ...
20
votes
4answers
677 views

Did Euclid prove that Pi is Constant?

Pi is defined the ratio of the circumference of a circle to its diameter, but of course different circles have different circumferences and diameters, so in order for it to be well-defined we need to ...
2
votes
1answer
110 views

How to cut circle into $n$ parts (all cuts are parallel to each other) so that each chunk is the same area (i.e. $\pi r^2/n$)?

I have been working this problem for a few hours today, but I'm stuck. I started working on a case where $n = 3$: Let the radius of the circle be centered at $(0,0)$, with a radius $r$. The equation ...
1
vote
1answer
142 views

Prove using integration "circle is a polygon when number of sides-> infinity

Is there a proof of "if number of sidesof a regular polygon ->infinitythe regular polygon -> circle." using integration?
0
votes
1answer
200 views

How to Find the First Moment of Area of a Circular Segment by Integration

Given a segment of circle symmetric about the $y$-axis, I'm wondering how to apply the integral $Q_x = \int y \, dA $ to find the first moment of area with respect to the $x$-axis. I'm having ...
0
votes
1answer
105 views

Bounds of double integral given a circle and a line

Calculate the double integral of the area between the function $$x^2+y^2=25$$ and the line $$y=-x+5$$ in the first quadrant. Now, I am unsure how to choose the bounds for y, I understand that the ...
1
vote
3answers
610 views

Center of Mass of a Circle

How would one find the center of mass of a circle? The center of mass of a rod is given by: $$\frac{1}{M}\int^{L}_{0}\rho x dx$$ So, for a sphere, it would be an area integral, such as: ...
1
vote
3answers
76 views

Another complex analysis question

I am going to have an analysis exam soon and I found the following question in a past paper: Evaluate the integral counterclockwise $$\int |z| \overline{z} \, dz$$ where y is the closed curve ...
1
vote
1answer
97 views

Analysis Exam Questions

I am going to have an analysis exam soon and I found the following question in a past paper: Evaluate $$\int \frac{-y \, dx + x \, dy}{x^2+y^2}$$ a) Once counterclockwise around the circle $$x^2 + ...
1
vote
2answers
216 views

Area of a circle

I've tried to find as a personnal exercise where the formula $A=\pi R^2$ comes from. After drawing the problem, I've found that $A = 2\int\limits_{-R}^{R}\sqrt{R^2-t^2}dt$. How can I calculate this ? ...
6
votes
3answers
1k views

Definite integral: $\displaystyle\int^{4}_0 (16-x^2)^{\frac{3}{2}} dx$

The following integral can be computed using the substitution $x = 4\sin\theta~$ and then proceeding with $dx = 4\cos\theta~ d\theta~$, and evaluating the integral of $\cos^4\theta~$: ...
0
votes
4answers
4k views

How do you find the formula for an area of the circle through integration?

I got this question on my Maths exam today, and the Department of Education has stated it is on the syllabus, but none of the three textbooks I could get my hands mention anything about it. One ...
2
votes
2answers
631 views

subvolume area under the intersection of a plane (line) and a cone

I am implementing a conical filter for Gupta-Sproull anti-aliased line algorithm. Given a cone with the total volume of 1 and a radius of 1. Find the subvolume of the intersection of a line. The ...
5
votes
4answers
1k views

Calculate $\pi$ precisely using integrals?

This is probably a very stupid question, but I just learned about integrals so I was wondering what happens if we calculate the integral of $\sqrt{1 - x^2}$ from $-1$ to $1$. We would get the surface ...