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♫Theme song♫
DORA: Hola, soy Dora. BOOTS: And I'm Boots! DORA: And today, we're going to LaSalle Mountain,
LaSalle Montaña! Can you say montaña? *Pause* Great!
Will you help me get to LaSalle Mountain before 8:00 am? *Pause*
Awesome! Let's go! (DORA and BOOTS run.)
♫Run, Dora, run! Corre, Dora, corre!♫
BOOTS: But, Dora, we don't know anything about finding volumes using calculus!
DORA: Who do we ask for help, when we don't know how to do something?
BOOTS: The textbook! DORA: The textbook; that's right!
Will you use the textbook to show us how to find volumes? *Turns around to show backpack*
You have to say textbook! BOOTS: *Enthusiastically* Say textbook!
TEXTBOOK: ♫ I'm the textbook. I'm the textbook.♫ OTHERS: ♫He's the textbook. He's the textbook.♫
TEXTBOOK: ♫ I'm the textbook!♫ Dora and Boots need to get a five quick! But they don't
know how to find volumes! Well, I know how to find volumes using integration. First,
you have to go through Washer Tunnel. *sound effect* Then, you go around Disk Fountain.
*sound effect* Finally, you have to walk across South Shell Lawn. *sound effect*
So remember, Washer, Disk, Shell! Say it with me. Washer, Disk, Shell. Washer, Disk, Shell. Washer, Disk, Shell!
Washer, Disk, Shell. Where do we go first? *Pause*
The Washer Tunnel! Right! *Runs a bit* Donde esta? Where is the Washer tunnel? *Pause* BOOTS: *Points* There it is.
♫Run, Dora, run! Corre, Dora, corre!♫ DORA: Look! We've made it to the tunnel! CHICO THE PIG: Ayudarme! Ayudarme!
BOOTS: Oh no, Chico. What's wrong? DORA: Oh no! CHICO: I'm trapped in this tunnel.
DORA: Chico needs to solve this calculus problem so that he can get out of this tunnel. Will you help us solve this calculus problem?
So, first, we graph this equation. You write the equations in y equals.
So this equation is y equals the square root of x while this one is y equals x.
So now we set them equal
to each other and we see that they
intersect at one. So we find the integral of this equation from zero
to one. So we write it in this format because this is a washer problem.
Because it revolves around the y and negative one.
So we add one to this so the equation on top, which is
square root of x plus one squared minus one plus x squared.
And we take the integral of this.
And then we multiply this out into this. And then simplify it.
And then we take the antiderivative of it. And then we plug in one which equals this.
Minus what we get after plugging in zero which equals that.
And the answer to this problem is pi over two. Whoohoo~
DORA: That's great! Now what is the answer? *Pause*
DORA: That's right! BOOTS: Phew. That was close.
Now Chico can finally escape this tunnel.
(CHICO escapes.)
BOOTS: Where do we go next? DORA: Let's see. Washer, Disk, Shell. Where do we go next? *Pause*
Disk Fountain! That's right! *Runs a bit* Where is Disk Fountain. Donde esta? *Pause*
BOOTS: *Points* There it is. DORA: Vamonos! Let's go.
♫Run, Dora, run! Corre, Dora, corre!♫
BOOTS: We made it to Disk Fountain. We need to get around the Disk Fountain really fast.
DORA: Really, really fast! BOOTS: Really, really, really fast!
DORA: Look! That calculus problem is heading straight towards us! BOOTS: It's not stopping! We need to
solve it quickly! DORA: We need a calculator to solve this
problem! My backpack might have a calculator! Will you ask backpack for me?
Say backpack! BOOTS: Backpack!
*Pause*
BACKPACK: ♫ Backpack x4. I'm the Backpack;
Loaded up with things and nick nacks too; Anything that you might need I got inside
for you. Backpack x4. YEAH!♫ Will a wallet help us solve this problem? Will a cell phone help us solve this problem?
Will a calculator help us with this problem?
Why yes! Calculator!
CALCULATOR: Find the volume of the solid whose
base is the region bounded between the curve y=x^3 and the y-axis from y=0 to y=1 and whose
cross sections taken perpendicular to the y-axis are squares.
PROBLEM: That's right. First,
graph the equation y equals x cubed.
Which from zero to one which is indicated
in the problem. So that's supposed to be a 3D drawing.
So, we rewrite y equals x cubed
as x equals the cubed root of y because
x is the base of the square. So we take the integral of the cube's root
of y squared. Because: find the area of each individual square on the interval.
So if we take the antiderivative of it, it equals
three y to the five thirds over five from one to zero.
Which equals three fifths when we plug in one. Minus zero.
So the answer to this problem is three fifths! Now where are you two going in such a hurry?
BOOTS: We have to go to LaSalle Mountain to get our fives. PROBLEM: Yay! Then you better hurry! Run, Dora, run! Corre, Dora corre!
DORA: We need to catch the stars! Reach up to catch the stars! Reach up! Catch 'em! Catch
'em! Catch 'em by finding the area between the curves! So first we graph y=cos2x.
And we have the lines x equals pi over 4 and x equals pi over 2.
So, to find this area, we take the absolute value of the integral of cosine two x from pi over four to pi over two.
So the reason that we are taking the absolute value is because the
area is below the x axis. Which means it is technically negative when we take the integral but the area should be positive. So, to do this on the calculator, first we press the button y equals, and put in cosine two x.
Then we press graph, second trace, the number seven. the lower limit is pi over four. Press enter. The upper limit is pi over four.Also press enter, and we get that the area is zero point five.
BOOTS: We're going to LaSalle Mountain to get our 5's! (Suddenly, a rock rolls in their way.)
DORA: Oh no! This rock is in our way to go to LaSalle mountain! BOOTS: Now, what do we do?! DORA: We need someone to help us solve this
math problem on this rock so that it will roll away!
INTEGRAL STAR: So first we draw the problem. The equation is one over x squared plus one, so we're finding the integral from zero to one. According to theorem seven point three point two (Volumes by Cylindrical Shells About the Y-axis), to find the volume,
we use the integral from zero to one of two pi x one over x squared plus one.
So, first, we press y equals in the calculator, and put in this equation. And then we push graph! Then we press second, trace, seven.
Put in zero as our lower limit. Press enter, and then one as our upper limit. And then press enter.
And the volume should be around 2.178.
Rounded to three significant digits.
Which is also equal to pi ln two.
DIEGO: Bye, Boots, bye Dora! BOOTS: Adios!
BOOTS: To stop Swiper, you need to help me! You need to say, "Swiper, no swiping!" Say it with me: Swiper, no swiping! Swiper, no swiping! Swiper, no swiping!
DORA: We did it. We did it. I can't believe we did it. Now, what was your favorite part of this episode? *Pause*