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.*

Quick Find Code For This Problem.

Legend: [QUES5LGND]

Solution For Part a.): [QUES5PA]

Solution For Part b.): [QUES5PB]

Solution For Part c.): [QUES5PC]

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**[QUES5LGND]**

**GREEN = CALCULATOR INSTRUCTIONS**

**BLUE = IMPORTANT NOTES/LINKS**

**RED = STUFF THAT NEEDS TO BE FOUND TO HELP SOLVE QUESTION**

**PURPLE = STEPS TAKEN**

The four permissible (TI-83) calculator functions that we're allowed to use to solve these wiki questions:

- Graphing the function: [Y=] [GRAPH]
- Determining the roots or y-intercepts: [2nd] [CALC] [2:zero]
- Numerically differentiating a function: [2nd] [6:dy/dx] or [MATH] [8:nDeriv]
- Numerically integrating a function: [2nd] [7:∫f(x)dx] or [MATH] [9:fnInt]

**[QUES5PA]**

**(a)** GIVEN FROM QUESTION ARE: AND

* Intersects* must be found so that they can be used as

Normally I would algebraically 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, it leads to a dead end.

Since this is a transcendental function, a function that contains an exponential function and a trigonometric function, we can’t apply the algebra we know to solve for the roots, so we have to use our calculator.

*Plug the two functions SEPARATELY into Y variables...in Y1 and Y2. Then press 2nd Calc 5 for the intersect function. Looking at the graph, we see R is bounded at x = 0 and at x = ???, where ??? is the x-coordinate of where the two functions intersect.*

Since the algebraic 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.

Using the calculator to determine the roots of e^(-x^2) + cos(x) - 1 = 0, I obtain x = 0.9419441. Although this isn't the only intercept, x = 0.9419441, we're only interested in the intercept that's close x = 1, because that's where the x-coordinate of the point of intersection is. We can see this on the graph as well.

e^(-x^2) + cos(x) - 1 = 0 at x = 0.9419441

Therefore...

Using fnInt function in our calculator, we obtain A = 0.59096245.

**[QUES5PB]**

**(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 * integrate the area 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.**It is important to visually see the actual shape, because it REALLY helps when revolving a 2D object and pretending as if it was three dimensional.

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

****To use the fnInt function on my calculator:**

**Put the equation in Y1****Quit to the home screen ["2ND" + "MODE"]**:**Between the brackets, separated with commas, put equation, variable, lower limit and upper limit. [VARS Y1, X, 0, 0.91419441] OR plug and chug everything into fnInt**

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

V= 5.454328983 units cubed

For more on the washer method and volumes of revolving objects, perhaps take a peek at this.

**[QUES5PC]**

**(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.

It's important to remember that the **area of a triangle is equal to 1/2 (base)(height).**

This is visual representation of what a cross section would look like. The * height* isn't given so in order to find it, some

In order to find the actual value of the height, you can either *use the pythagorean theorem or even any of the trigonometric ratios that can be used to find the side of a right triangle.*

A side note is that we know all **equilateral triangles have sides that are equal in length as well as the size angles. The angles in equilateral triangles are all equal to .**

The height is positioned opposite the angle. Therefore, it is true to say that:

Since the * sine of an angle is opposite/hypotenuse. h is opposite the angle and the expression is the hypotenuse*.

Now that the height is found and the other variables needed to find the volume of the cross section are given, *plug in the values known as the base, height and width for this question.*

V = 0.9387 units cubed.

BOOMDEYADA~!