# Semicircle problem

1. Sep 18, 2005

### n05tr4d4177u5

i have been told that dA=R dR d(theta), what does mean exactly in terms of pi. the shape is a semicircle. Does this mean that d(theta) is 180, or pi. Please give me an example.

2. Sep 18, 2005

### arildno

No, dA is a notation for an area element (infinitely small).

1. Suppose you are using polar coordinates $$r,\theta$$, what sort of region is described by the inequalities:
$$R\leq{r}\leq{R}+\bigtriangleup{R},\theta_{0}\leq\theta\leq\theta_{0}+\bigtriangleup{\theta}$$

You ought to see that this represent a specific portion of a section of a circle.

2. Now, if you let $$\bigtriangleup{R},\bigtriangleup\theta$$ tend towards zero, you may approximate this region as a rectangle (or trapezoid, if you like that better). What should the (tiny!) area of this figure be?

3. Sep 18, 2005

### n05tr4d4177u5

that doesnt really help my question

here is an example of mine that i have
suppose i hae a rectangle,

in a rectangles case, dA = dX dY

if i have the rectangles width and it is 4 m, then dA = 4dY,

which lets me integrate along y axis,

now for a semicircle, how would i do that if dA = r dr d(theta)

4. Sep 18, 2005

### n05tr4d4177u5

also

so what i did, was changed dr to dy, and now i need to change d(theta) which should be a number (like in the case of 4 m for rectangle), and it has to be in terms of pi,
the shape is a semi circle

5. Sep 18, 2005

### arildno

You do know how to integrate, don't you?
If you are to compute the area of the semicircle, the $$\theta$$ integration is trivial, so you can find out that by yourself.

6. Sep 18, 2005

### n05tr4d4177u5

i dont know

no i dont know, its been a while since i integrated a circle

please help me, thats why im asking, i havent done that in years

7. Sep 18, 2005

### arildno

Well, can you at least post the integral you want to compute, specifying the intervals in which R and the angle must lie in for the semi-circle?

8. Sep 18, 2005

### n05tr4d4177u5

y = d(theta) Y ^ 2

integrate from y=R to y= 0

Shape is semi circle

9. Sep 18, 2005

### arildno

This is not what is in your book (it doesn't make any sense). Please type in correctly.

10. Sep 18, 2005

### n05tr4d4177u5

imagine the top half of a circle. the origin lies along the bottom of the semicircle, and in the middle. y axis up, and x axis to the right and left. i think theta can only go from 0 to 180 degrees since it is a semi circle. Y = d(theta) R squared
R = radius, integrate from 0 to R

11. Sep 18, 2005

### arildno

What is Y?
Do you know what the d in front of (theta) means?

12. Sep 18, 2005

### n05tr4d4177u5

in a rectangles case, dA = dX dY

if i have the rectangles width and it is 4 m, then dA = 4dY,

which lets me integrate along y axis,

now for a semicircle, how would i do that if dA = r dr d(theta)

13. Sep 18, 2005

### arildno

Well, why do you believe this to be true?

How do you derive that?

14. Sep 18, 2005

### n05tr4d4177u5

lol

are u reallly an advisor, cuase its unbelievable youve never seen that before,

15. Sep 18, 2005

### n05tr4d4177u5

example , the integral of ydy = y squared/ 2

16. Sep 18, 2005

### arildno

Sure I have, but you are so vague and obtuse that I don't think you understand what you write yourself.
So, again:
show that you have at least a minimum of mathematical skill and understanding.
I have already shown you how you can do this problem, but it doesn't seem you have the competence to follow a perfectly clear line of reasoning.

17. Sep 18, 2005

### Integral

Staff Emeritus
You are good up to here.
Why can you write this? Where did it come from? You cannot just drop the dx and it is not automaticlly set to 4 or any other value.

One of the first things you need to do is carefully define your rectangle in terms of your coordinate system.

Suppose one corner is at the origin, x=y=0 and the diagonal corner is at x=L, y=W. To find the area we must define our integral:
$$\int dA = \int^{y=W}_{y=0} \int^{x=L}_{x=0} dx dy$$
Can you complete the above integral?
Ok, now lets set up the integral for the area of a semi circle.
Once again we need to define the circle in terms of the coordinate system.

If the center of the circle is at the origin and the radius is R.

So to get the Area of the circle (or portion there of) the radius will take on values between 0 and R, while the angle $\Theta$ varies from 0 to $\pi$ (for a semi circle). Now lets set up the integral

$$\int dA = \int ^{\Theta=\pi} _{\Theta=0} \int^{r=R}_{r=0} rdr d\Theta$$

Can you evaluate the integral?

18. Sep 18, 2005

### HallsofIvy

Staff Emeritus
You haven't stated a problem here! What do you want to integrate from 0 to R? What function are you integrating?

If you just want the area, look at the problem you gave:
(And arildno did not say he had never seen that before, he was trying to get you to think about why you think that was true.)

Suppose your semi-circles radius is R. Then dA= r dr dθ. In a sense, dr here, as well as r, is r: along each radius we go from r= 0 to R so the "change in r", dr, is R.

Okay, then r dr dθ becomes dA= R2 dθ, just like your dX dY became 4dy, the area of the semi-circle is
$$\int_{\theta=0}^{\pi} R^2 d\theta= \pi R^2$$, exactly the right answer.

(θ does not go from 0 to 180! As I am sure you learned in when you were learning the derivatives of sine and cosine, in calculus, all angles are in radians.)

(Oh, and by the way, why was this posted under "differential equations". Was this part of a differential equations problem?)

Last edited: Sep 18, 2005
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