MATH3503 Online Assignment 201

• 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 4 significant digits (e.g., 12.34, 0.01234, etc.) in your final answer, and a couple of more digits in the intermediate steps.
• Arithmetic operators:

  Multiplication	* (You cannot omit it!)
  Division	/

• Available functions
  • Trigonometric functions: sin(x), cos(x), tan(x), sec(x)
  • Exponential and logarithmic functions with base e: exp(x) (`=e^x`), ln(x) (or log(x))
  • Square root: sqrt(x)
  • Absolute value: abs(x)

Midterm 1. October 22, 2020

1. Compute the following integrals.
   1.   `int_{0}^{1} 4/(6x+9)dx =` Enter a numerical value with more than 4 significant digits.
   2.   `int_{-4//2}^{4//2} 9/sqrt{4 - 2x} dx =` Enter a numerical value with more than 4 significant digits.
      
   3.   \(\displaystyle\int (3x+4)^{-3/5} dx =\)`+C`
      *Enter `x^{-a//b}` as x^(-a/b).
   4.   `int dx/(cos^2(4x)) =` `+C`
      *Hint: `1/{cos theta} = sec theta`.
   5.   `int sin x sqrt{cos^{5}x}\ dx =``+C`
      * Enter `cos^{a//b}x` as cos(x)^(a/b).
   6.   `int dx/(x ln(8x)) =``+C`
      
   7.   `int x e^{-7x}dx=``+C`
      *Enter `e^x` as exp(x).
   8.   `int x^{3}cos(5x^{2})dx =``+C`
      

   
   
2.  
   Compute the area surrounded by `y = x^{2.5}` (blue) and `y = x^{1//2.1}` (red) ` =`
   

   
   
   
   
3.  
   Waste water is flowing into a tailings pond at the rate of `r(t) = 3(1+t)^{-1.4}`.

   1. Find the total volume `V(t)` of the water in the tailings pond at time `t`, given that `V(0)= 3`.
      ` V(t) = `
   2. After a long time what will be the volume of water? I.e., `V(\infty) = `
      *Hint: `infty+`constant `=infty`,  \(1/\infty = 0\)
   
   
4.  
   The accelerometer on board a dump truck recorded its vertical acceleration while unloading dirt; \[ a(t) = 4e^{-t/2}\cos(5t). \] Find its vertical velocity `v(t)` if `v(0) = 0`. [Hint: `a={dv}/{dt}`] `v(t) = `
   
   
5.  
   You are designing an open pit whose horizontal cross-section is square with each side equal to \(\displaystyle L(y) = \frac{75}{\sqrt{1+y/3}}\)[m] at depth `y` [m]. If the deepest part of the pit is `y=9` [m], find the total volume of dirt that needs to be removed. Volume of dirt = [m³]
   
   
6.  
   A mineral deposit was found to have the shape of a solid of revolution produced by rotating \[ \frac{x^2}{36}+\frac{y^2}{49} = 1, ~~~-6\le x \le 6~~\text{and}~~-7\le y \le 7\] about the `y`-axis. Find its volume. [Hint: it is an ellipse centered at the origin.]
   Volume =

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