4
votes
2answers
652 views

The area of circle

The question is to prove that area of a circle with radius $r$ is $\pi r^2$ using integral. I tried to write $$A=\int\limits_{-r}^{r}2\sqrt{r^2-x^2}\ dx$$ but I don't know what to do next.
3
votes
4answers
105 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
63 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
52 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
81 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
185 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
117 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 ...
22
votes
5answers
822 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
117 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
2answers
177 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
211 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
121 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
672 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
78 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
219 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
2k 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
5k 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
642 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 ...