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Question 5

This version was saved 14 years, 11 months ago View current version     Page history
Saved by Jamie
on April 19, 2009 at 7:58:38 pm
 

A Calculator MAY be used to help answer this question.

Be certain to limit yourself to the four permissable calculator functions.

 

Let R be the shaded region in the first quadrant enclosed by the graphs of , , and the y-axis, as shown in the figure at right.

 

(a) Find the area of region R.

 

 

(b) Find the volume of the solid generated when the region R is revolved about the  line y = -1.

 

 

(c) The region R is the base of a solid. For this solid, each cross section perpendicular to the x-axis is an equilateral triangle. Find the volume of this solid.

 

 

 

 

 

Solution

 

(a) GIVEN FROM QUESTION ARE: AND

 

Intersects must be found so that they can be used as limits for the integral, or from which points the area of the region shall be calculated. In order to do that;

 

Normally I would manually solve for the intersects by making the two equations equal to each other and solve for x. But because e to the negative x squared is such a monster to deal with, I think it leads to a dead end.

 

Since the preferrable way to find intersects by making them equal to each other isn't working, I also tried massaging it a little by moving the terms to one side and finding the roots.

 

I couldn't quite do that either, so I'm sorry that I cheated and used the intersect function on my calculator. Plug the two functions SEPARATELY into Y variables.. in Y1 and Y2. Then press 2nd Calc 5 for the intersect function. You may have to use this twice since there are probably two intersects.

 

As a result, the calculator gave me a rough estimate of intercepts where x = - 0.9419441, 0.9419441

 

Therefore...

 

Once this is evaluated, the sum of the area in between the curve ends up being A = 2.0528878 => 2.0529

 

(b) To find volume there is usually a general formula. I don't like using that term now. I prefer the word, PATTERN.

 

When an area is being rotated around the x-axis, the general formula is:

Because the area is revolving around the axis, it forms a washer shape. To get the volume, we get the area that is given. R is the bigger radius, and is the upper  function and could also be looked at as f(x) and the lower function, g(x) is the smaller radius. This lower function forms the "hole" in the washer.

 

For visual representation, refer to this diagram, which is looking at the function from a different side; a different perspective.

 

 

 

The upper function in this problem is . The lower function or radius is . Therefore

 

 

 

The reason why each radius has a "+1" added, is because the area is not being revolved exactly on the axis, but rather, one unit below the axis, therefore from each radius, -1 is subtracted. Two negatives of course makes a postive. This can be simplified to:

 

Usually I  would antidifferentiate the monster of an equation and plug in the limits to find the integral to subtract, but I don't think seems it can be antidifferentiated, I'm just going to plug this equation into my calculator and use the "fnInt" function to integrate from -0.91419441 to 0.91419441.

 

**To use the fnInt function on my calculator:

  1. Put the equation in Y1
  2. Quit to the home screen ["2ND" + "MODE"]
  3. Between the brackets, separated with commas, put equation, variable, lower limit and upper limit. [VARS Y1, X, -0.91419441, 0.91419441]

 

Then the sum will be found for you. This leaves us with: 

(c) Firstly, the question states that the shaded region is the base of the solid. One face of the solid is said to be the shape of an equilateral triangle, as are the other faces of each cross section.

 

The area of any triangle is

jkgkh

 

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