1. ## Cylinder

Let r be a positive constant. Consider the cylinder $\displaystyle x^2+y^2≤r^2$ and let C be the part of the cylinder that satisfies 0 ≤ z ≤ y.

(3) Let a be the length of the arc along the base circle of C from the point (r, 0, 0) to the point (r cos θ, r sin θ, 0) (0 ≤ θ ≤ π). Let b be the length of the line segment from the point (r cos θ, r sin θ, 0) to the point (r cos θ, r sin θ, r sin θ). Express a and b in terms of r, θ.

( 4) Calculate the area of the side of C with x2+y2 = r2, and express it in terms of r. 2. Relevant equations Not sure 3. The attempt at a solution

I used the formula . [f '(x)]² = x²/(r²-x²) ... r∫π0dθ=??? the answer ir θr

3. ## Re: Cylinder

$$x^2+y^2=r^2$$

$$2x + 2y \dfrac{dy}{dx} = 0$$

$$\dfrac{dy}{dx} = -\dfrac{x}{y}$$

$$\dfrac{dy}{dx} = \dfrac{x^2}{y^2}$$

So, your formula is $L = \int_r^{r\cos \theta} \sqrt{1+\dfrac{x^2}{y^2}}dx$

Next, we have $x = r\cos \theta, dx = -r\sin \theta d\theta, y = r\sin \theta$, and at $x=r$, $\theta = 0$ and at $x=r\cos \theta$, $\theta = \theta$

$$L = \int_r^{r\cos \theta} \sqrt{1+\dfrac{x^2}{y^2}} dx = r\int_0^\theta \sqrt{1+\cot^2 \theta} \sin \theta d\theta = r\int_0^\theta d\theta = r\theta$$

4. ## Re: Cylinder

thanks a lot!!!!

why you substituted dx=−rsinθdθ for dx=rsinθdθ? The -r changed for +r?

5. ## Re: Cylinder

Originally Posted by helpprovaexata
thanks a lot!!!!

why you substituted dx=−rsinθdθ for dx=rsinθdθ? The -r changed for +r?
I typed it from my phone. Arc length is always positive. I missed the absolute value signs.

6. ## Re: Cylinder

in the second part the z-axis varied rsinθ so just put this?

the answer is rsinθ

7. ## Re: Cylinder

Originally Posted by helpprovaexata
in the second part the z-axis varied rsinθ so just put this?

the answer is rsinθ
Correct

8. ## Re: Cylinder

How would i handle the integral of question d?

9. ## Re: Cylinder

??What "question d"? Do you mean question 4, "Calculate the area of the side of C with $x^2+y^2 = r^2$"?

10. ## Re: Cylinder

Yes, number 4. Sorry

11. ## Re: Cylinder

Basically, in part 3, we found a map from $(x,y,z)$ to $(r, \theta)$. The base of the cylinder is $r\theta$ while the height is $r\sin \theta$. You have $0< \theta < \pi$ (because for the rest, $z=0$). So, you are taking the integral $\displaystyle \int_0^\pi r^2\sin \theta d\theta = 2r^2$.