MATH3503 Online Assignment 4

• 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.

Area between curves and Volume

1.  

   The area surrounded by `y = x^{2.3}` (blue) and `y = x^{1/2.5}` (red) ` =`

   
   
2.  

   The area surrounded by `y = (x-5)^2+1` (blue) and `y = 3x` (red) ` =`

   
   
3.  

   The area surrounded by `y = x^2-2` (blue) and `y = -2x` (red) and the x-axis ` =`

   
   
4.  
   A certain smelting furnace has the power input of `P_\text{in} = 9+sin(5t)`[kW], and heat escapes at the rate of `P_\text{out}=7+sin(5t + 0.6)`[kW]. Find the increase in the heat [kJ] in the interval \(0\le t\le 1\) [sec].
   Increase in heat =
   
   
5.  
   You are designing a circular open pit that has a cross-section of radius \(\displaystyle r(y) = \frac{84}{1+y/8}\)[m] at depth `y` [m]. If the deepest part of the pit is `y=24` [m], find the total volume of dirt that needs to be removed.
   Volume of dirt = [m³]
   
   
6.  
   Find the volume of a mound of dirt whose height at point `(x,y)` is defined by \[ z = -9x(x-3y), ~~~ 0\le x\le 3y,~~ 0\le y\le 5 \] Area of the cross-section at a given `y`: `A(y) = `
   *Hint: Think of `y` as a constant when integrating with respect to `x`.
   Volume =
   
   
   

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