THE QUEST TO BECOME A

MATHEMON MASTER!Jenny vs. Jane Battle: Solids of Revolution

Question

Rival Mathemon trainers, Jane and Jenny, are trapped in the forest and surrounded by wild Mathemon agitated by Jenny’s constant disturbance of peace through a series of attacks. Jane is a witness of this abuse and she does her best to protect the wild Mathemon by challenging Jenny to a battle to stop her vicious ways.

.:. THE SITUATION .:.

You have challenged Rival Jenny!

Jenny sent out Revoloo!

“Go! Gralinte!!”

EPIC RIVAL BATTLE!!

Jenny used a DIAGRAM!

Let R be the shaded region bounded by the graphs of y = , y = and the vertical line x=1.

GO GET ‘EM, REVOLOO!

xe 2x6

The foe’s Revoloo’s DEF has risen!

Jane used an INT BOOST!

Gralinte’s intelligence has risen!

EPIC RIVAL BATTLE…

The foe’s Revoloo used QUESTION!!

(a) Find the area of the enclosed region R.

It’s a critical hit! Gralinte takes 12 damage!

QUESTION…BATTLE

Gralinte used POINTS OF INTERSECTION!

One of the definitions of integration can be known as finding the area underneath a curve. In this case, R is the area that needs to be integrated.

But since R is not infinitely continuous and occurs on a closed interval, the points of intersection must be found in order for the area to be integrated.

SOLUTION…BATTLE

To find the points of intersection, make both functions equal to each other and find the values 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 is integrated to 1, since it was given that it is enclosed by the vertical line x=1.

SOLUTION…BATTLE

The point of intersection occur at approx. x=0.1081. But since that number is quite complex, we can assign a letter to that number. So we can let T=0.1081.

The foe’s Revoloo took 9 damage! Revoloo is paralyzed!

SOLUTION…BATTLE

Revoloo took 6 damage!

SOLUTION…BATTLEGralinte used INTEGRATION!

Now knowing the intersections, we can now use those as our interval from which we are going to integrate and find the area for. Remember that we let T = 0.1081 since we wouldn’t want to waste our time rewriting such complex numbers all the time. Solving for this using our calculators we should get….

A≈1.2398 units²

dt

“Hmph, you think you’re so tough Janey Poo? See if you can handle this next attack! Go get’em Revoloo!”

Revoloo used QUESTION!

(b) Find the volume of the solid generated when R is revolved about the x-axis.

It’s a critical hit! Gralinte takes 17 damage!

QUESTION…BATTLE

Gralinte used INTEGRATION!

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 revolve shapes 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 is πr2. This is because the cross sections of the area itself is in the shape of a circle as it revolves around the specific axis of rotation.

SOLUTION…BATTLE

Since the area is consisted of two different curves, the cross section looks like a washer. The upper function will be the bigger radius R, and the lower function will be the smaller radius, r. As seen here:

SOLUTIONS…BATTLE

In general, we should know that to find the volume of a washer type question, we will need to take the area of the large circle and the area of the small circle and then subtract them from each other. From there, just integrate along the interval, or through the intersections.

SOLUTION…BATTLE

dx

ITS SUPER EFFECTIVE! Revoloo takes 24 damage!

Revoloo used DUST!

Gralinte has been blinded! Gralinte takes 2 damage!

“Gralinte, that’s enough! Come back!Go, Derivee!”

COUNTER ATTACK!

Derivee used ANTI DIFFERENTIATE!

SOLUTION…BATTLE

Before we begin anti differentiating, let’s first simplify the integral to make it easier.

dx

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

Revoloo took 12 damage!

SOLUTION…BATTLE

3units8073.2V

We can now finish the question by applying our knowledge of the fundamental theorem of calculus, which is that the integral of a derivative = total change in the parent function along the same interval. Previously, we found the parent function so now all that’s left is to solve. Our approximate answer should be:

“You think you’re so great? There’s no way I’d let you win! Wasn’t it obvious that I was just playing around? Well, no more fooling around! Take this!”

Revoloo used QUESTION!

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

Derivee is crushed by the sudden overwhelming attack! Derivee takes 19 damage!

QUESTION…BATTLE

Derivee used LIST!

Revoloo takes 10 damage! Revoloo is stunned and is in critical condition!

SOLUTION…BATTLE

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

Lheight 5 xexL 26 Like the previous part, we should also be able to find the volume of the solid by taking the area of said solid and integrating it. The interval will remain the same since that is where the actual solid starts and ends.

lwhA

“Derivee, that’s enough! Come back!

Go! Gralinte!!”

Gralinte used INTEGRATION!

SOLUTION…BATTLE

As stated earlier, to find the volume all we have to do is integrate the area of the rectangle. Let’s first take a look at the rectangle so that we know how to substitute in our values.

SOLUTION…BATTLEThe width, as you can see, is basically the infinitely small values of “x” as you go along the interval. Since it is so small, we just leave it as “dx”.

Now knowing all our values, we can now integrate:

Then solve:

The foe’s Revoloo has fainted! Gralinte gained 42 EXP. Points! Gralinte grew to LV 4!

DEFEAT!!!