jenny vs. jane

8/14/2019 Jenny vs. Jane

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Jenny vs. Jane Battle: Solids of RevolutionQuestion


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Rival Mathemon trainers, Jane and Jenny,are trapped in the forest and surrounded by

wild Mathemon agitated by Jennys constantdisturbance of peace through a series ofattacks. Jane is a witness of this abuse andshe does her best to protect the wild

Mathemon by challenging Jenny to a battleto stop her vicious ways.


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You have challenged Rival Jenny!

Jenny sent out Revoloo!

Go! Gralinte!!


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Jenny used a DIAGRAM!

LetR be the shaded region bounded by the graphs of

y = , y = and the vertical line x=1.x

e2

x6

The foes Revoloos DEF hasrisen!


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Jane used an INT BOOST!

Gralintes intelligence has risen!


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Gralinte used POINTS OF INTERSECTION!

One of the definitions of integration can beknown as finding the area underneath a

curve. In this case, R is the area that needsto be integrated.

But since R is not infinitely continuous and

occurs on a closed interval, the points ofintersectionmust be found in order for thearea to be integrated.


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To find the points of intersection, make bothfunctions equal to each other and find thevalues where these functions meet.

There are two ways that you can do this.Manually, you can find the value(s) of x, or

you can use the intersect function on your

calculator. In this specific case, it isintegrated to 1, since it was given that it isenclosed by the vertical line x=1.


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The point of intersection occur at approx.

x=0.1081. But since that number is quite

complex, we can assign a letter to thatnumber. So we can let T=0.1081.

The foes Revoloo took 9 damage!Revoloo is paralyzed!


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Revoloo took 6 damage!

Gralinte used INTEGRATION!

Now knowing the intersections, we can now use those as ourinterval from which we are going to integrate and find the areafor. Remember that we let T = 0.1081 since we wouldnt wantto waste our time rewriting such complex numbers all the time.Solving for this using our calculators we should get.

A1.2398 units

d

t


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Hmph, you think youre so toughJaney

Poo? See if you can handle this next attack!Go getem Revoloo!


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Revoloo used QUESTION!

(b) Find the volume of the solid

generated when R is revolved aboutthe x-axis.

Its a critical hit! Gralinte takes 17damage!


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In every math problem lies a pattern. In this case, when we

revolve something around the x-axis or a horizontal line,

the pattern is in the general formula of how to revolveshapes around this axis.

The volume for revolution around a horizontal line or the x-axis is equal to the integral of the area of a circle, which isr2. This is because the cross sections of the area itself is inthe shape of a circle as it revolves around the specific axisof rotation.


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Since the area is consisted oftwo different curves, the cross

section looks like a washer.The upper function will be thebigger radius R, and the lowerfunction will be the smallerradius, r. As seen here:


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In general, we should know that to find the volume of awasher type question, we will need to take the area ofthe large circle and the area of the small circle and thensubtract them from each other. From there, justintegrate along the interval, or through the

intersections.

dx

ITS SUPER EFFECTIVE! Revoloo takes 24damage!


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Revoloo used DUST!

Gralinte has been blinded! Gralinte takes 2

damage!

Gralinte, thats enough! Come back!Go, Derivee!


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Derivee used ANTI DIFFERENTIATE!

Before we begin anti differentiating, lets first simplify theintegral to make it easier.

dx

From here, we can use simple anti differentiation rules to antidifferentiate.


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Revoloo took 12 damage!

3units8073.2V

We can now finish the question by applying our knowledge ofthe fundamental theorem of calculus, which is that theintegral of a derivative = total change in the parent functionalong the same interval. Previously, we found the parentfunction so now all thats left is to solve. Our approximateanswer should be:


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You think youre so great? Theres no wayId let you win! Wasnt it obvious that I was

just playing around? Well, no more foolingaround! Take this!


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Revoloo used QUESTION!

(c)The region R is the base of a solid. Forthis solid, each cross section perpendicularto the x-axis is a rectangle whose height is 5times the length of its base in region R. Findthe volume of this solid.

Derivee is crushed by the suddenoverwhelming attack! Derivee takes 19damage!


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Derivee used LIST!

Revoloo takes 10 damage! Revoloo is stunned

and is in critical condition!

The question has already given us two key things which isthe base and height of the rectangle as shown here:

Lheight 5=x

exL2

6

=

Like the previous part, we should also be able to find thevolume of the solid by taking the area of said solid andintegrating it. The interval will remain the same since thatis where the actual solid starts and ends.

lwhA =


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Derivee, thats enough! Come

back!Go! Gralinte!!


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As stated earlier, to find the volume all we have to do isintegrate the area of the rectangle. Lets first take a look atthe rectangle so that we know how to substitute in our

values.


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The width, as you can see, is basically the infinitely small values ofx as you go along the interval. Since it is so small, we just leaveit as dx.

Now knowing all our values, we can now integrate:

Then solve:


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The foes Revoloo has fainted! Gralinte gained 42EXP. Points! Gralinte grew to LV 4!

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