MATH3503 Online Assignment 9

• Make sure to use the link on the Learning Hub, which contains a user name and a user ID unique to you, otherwise your mark would not be recorded. You may use the URL without '?uname=...&uid=...' if you just want to practice. You will get different numbers every time you open the page.
• Numerical answers are evaluated as correct if the difference from the correct value is less than `0.001 times` (correct value) unless otherwise specified. That means, you need to keep at least 5 significant digits (e.g., 12.34, 0.01234, etc.) in your final answer, and a couple of more digits in the intermediate steps.

Arclength

1.  
   \(\displaystyle y = \frac{2}{3}(x^2+1)^{3/2}\) in interval `1\le x\le 4`.
   Arclength =
   * Hint: \( 4a^2 + 4a + 1 = (2a+1)^2\)
   
   
2.  
   \(\displaystyle y = \frac{1}{4x^2}+\frac{x^4}{8}\) in interval `4\le x\le 5`.
   Arclength =
   * Hint: Express inside the square root as a single fraction, then use \( a^2 + 2a + 1 = (a+1)^2\)
   
   
3.  
   \(\displaystyle y = \frac{e^{3x}+e^{-3x}}{6}\) in interval `-7\le x\le 7`.
   Arclength =
   
   
4.  
   \(\displaystyle y = \ln(\sec x)\) in interval `-\pi/7\le x\le \pi/7`.
   Arclength =
   * Hint: \(\int \sec x\, dx = \ln(\tan x + \sec x) +\) constant.
   
   

   Surface Area

   Compute the surface area of each of the following solids of revolution in the given interval.
5.  
   \(\displaystyle y = 8x\) rotated about the `y`-axis in interval `0\le y\le 4`.
   Surface area =
   
   
6.  
   \(\displaystyle y = \sqrt{7x}\) rotated about the `x`-axis in interval `0\le x\le 5`.
   Surface area =
   
   
7.  
   \(\displaystyle x^{2/3}+y^{2/3} = 25\) rotated about the `x`-axis in interval `0\le x\le 125`.
   Surface area =
   
   
8.  
   An open pit has a paraboloid shape. Its height is 44 metres and the diameter of the opening is 165 metres. Compute the surface area of this pit.

   The paraboloid can be created by rotating

   `y=``x^2`   (Keep a few more significant digits.) about the `y`-axis. Thus, Surface area = m²

   
   

If you are ready to submit the result, click Submit!   *Make sure that all questions are evaluated. Click "Check answers!" before clicking "Submit!".