# axis

1. ### Finding the angle and axis lengths of an ellipse

Hi, I have an ellipse defined by \|\mathbf{A}^{-1}v\| = 1, where v is a vector in the ellipse and \mathbf A is a real, invertible 2-by-2 matrix. How can I find the lengths of the axis of the ellipse as well as the angle of the major axis? I need to know this because I want to plot the ellipse...
2. ### Second Moment of Area about x-x Centroidal Axis

Having problems with (b) and (f) here. For (b) I've done the following: A1 = 600 mm^2 y1 = 50 mm h1 = 27.78 mm A2 = 1000 mm^2 y2 = 30 mm h2 = 7.78 mm A3 = 2000 mm^2 y3 =10 mm h3 = 12.22 mm I = bd^3/12 + Ah^2 = (30 * 20^3/12 + 60 * 27.78^2) + (50*20^3/12 + 1000 * 7.78^2) + (100*20^3/12 +...
3. ### find the area of surface generated by revolving the are about y axis

the question is x = (e^t)(cos t ) , y = (e^t)(sin t ) , about y axis , where y = between 0 and 1 i have problem of finding the value of t when 1 = (e^t)(cos t ) and also 0 = (e^t)(cos t ) , since the integral cant be integrate so i let is as I . how to continue form my working ?
4. ### find the area of surface generated by revolving the curve about y -axis

i am stucked here , how to continue ?
5. ### Finding the axis rotation of a quaternion

- How can i find the single axis rotation of a quaternion ? - How can i find the single axis rotation of a quaternion , followed by another single axis rotation - Calculate the new co-ordinates of a point after a single axis rotation
6. ### Projection Transformation on x Axis Parallel to y=2x

Let $T \colon \mathbb{R}^2 \rightarrow \mathbb{R}^2$ be the projection transformation on x axis in parallel to $y=2x$. Find the representing matrix of $T^*$ for the standard basis $B=\left(\left \{1,0 \right \},\left\{0,1\right\}\right)$. I have a solution which I don't understand. A basis for...
7. ### How to evaluate solid angle subtended by an ellipse at any point on the vertical axis

How to evaluate solid angle subtended by an ellipse at any point on the vertical axis passing through its center, I have made some calculations for this that is shown by the image attached
8. ### about the x or y axis

If your doing disk, washer, cyndrical shells, and the question says (about y = "some positive number") or (about y = "some negative number") or (about x = "some positive number") or (about x = "some negative number") then how do we put that in the formula? (for disk method) Ex. about x =...
9. ### Shell Method Problem - about x axis

y = x^{3} y = 8 x = 0 about x axis Converted: x = y^{1/3} y = 8 x = 0 Find limits of integration y^{1/3} = 0 y = 0 Problem: V = 2\pi \int_{0}^{8} (y)(y^{1/3}) dy V = 2\pi \int_{0}^{8} y^{4/3} dy V = (2\pi) (\dfrac{3}{7})y^{7/3} evaluated at 0...
10. ### Shell Method Problem - about x axis - # 2

Find Volume about x axis x = 5 + (y-6)^{2} x = 14, Simplify x = 5 + (y-6)^{2} x = 5 + (y-6)(y-6) x = 5 + y^{2} - 6y-6y + 36 x = 5 + y^{2} - 12y + 36 + 5...
11. ### Shell Method Problem - about y axis - # 6

y = 11(x)^{1/2}, y = 0, x = 1 about x = -3 so revolves about y axis. Find limit of integration 11x^{1/2} = 0 x = 0 V = 2 \pi \int_{0}^{1} (x)(11x^{1/2}) dx V = 2 \pi \int_{0}^{1} 11x^{3/2}) dx (2\pi) (\dfrac{2}{5}) 11x^{5/2} evaluated at 0 and 1 (2\pi) [[(\dfrac{2}{5}) 11(1)^{5/2}] -...
12. ### Shell Method Problem - about y axis - # 5

y = 3x^{4}, y = 0, x = 2 about x = 4 so revolves about y axis Find limit of integration 3x^{4} = 0 x = 0 V = 2\pi \int_{0}^{2} (x)(3x^{4}) dx V = 2\pi \int_{0}^{2} 3x^{5}) dx V = (2\pi) \dfrac{3x^{6}}{6} evaluated at 0 and 2 V = (2\pi) \dfrac{x^{6}}{2} evaluated at 0 and 2 V = (2\pi)...
13. ### Shell Method Problem - about y axis - # 4

y = 3x^{2} y = 18x - 6x^{2} about y axis Find Volume: Find limits of integration: 3x^{2} = 18x - 6x^{2} 3x^{2} + 6x^{2} - 18 = 0 9x^{2} - 18 = 0 9(x^{2} - 2) = 0 x = \pm \sqrt{2} V = 2\pi\int_{\sqrt{-2}}^{\sqrt{2}}[ (x) (18x - 6x^{2} - 3x^{2})] V = 2\pi\int_{\sqrt{-2}}^{\sqrt{2}}[...
14. ### Shell Method Problem - about y axis - # 3

y = (16x)^{1/2} y = \dfrac{x^{2}}{16} about y axis Find volume via shell method. Find limits of integration: (16x)^{1/2} = \dfrac{x^{2}}{16} How do you isolate x and get two limits of integration in this case?
15. ### Shell Method Problem - about y axis - # 2

Should the limits be changed, or can an answer be given without doing so? Show the answer is 11\pi(1 - \dfrac{1}{e}) y = 11e^{-x^{2}} y = 0 x = 0 x = 1 V = 2\pi \int_{0}^{1} (x)(11e^{-x^{2}}) dx u = -x^{2} du = -2x dx -\dfrac{1}{2}du = x dx V = 2\pi \int_{0}^{1}- \dfrac{1}{2}11e^{u}...
16. ### Shell Method Problem - about y axis

y = 5x(x - 1)^{2} y = 0 This line should say y = 0, not what written. x = 1 x = 0 Using shell method about y axis, find volume. V = 2\pi \int_{0}^{1} (x)(5x)(x - 1)^{2} dx V = 2\pi \int_{0}^{1} (x)(5x)(x - 1)(x - 1) dx V = 2\pi \int_{0}^{1} (x)(5x)(x^{2} - 2x + 1) dx V = 2\pi...
17. ### reflecting across an axis

Hi; Simple question just need a definitive answer do the two following statements mean the same thing? Reflect across the x-axis. Reflect over the a-axis. Thanks.
18. ### Shell Method Problem - about x axis

Example Problem (not the one solved but a model) The region bounded by the curve y = \sqrt{x}, y = 0, x = 4, revolved around the x axis. V = \int_{0}^{2} 2\pi(y)(4 - y^{2}) dy V = \int_{0}^{2} 2\pi(y)(4y - y^{3}) dy 2\pi[2y^{2} - \dfrac{y^{4}}{4}] evaluated at 0 and 2 is 8\pi Now the...
19. ### Calculate the surface area enclosed by y=4-x² and the x axis between 0≤ x ≤2?

Calculus assignment question Centroid?
20. ### Points on a cirlce to a line segment

Hello, Let say I have a circle with a known circumference that has arbitrary points on it around the circle. I want to convert/unravel that circle to a line segment and know where the arbitrary points on the circle reside on the line segment. So for example, if a circle with a circumference...