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Sketch the region enclosed by the curves and find its area.$$y=x^{2}, y=\sqrt{x}, x=\frac{1}{4}, x=1$$

$$\frac{49}{192}$$

Calculus 1 / AB

Calculus 2 / BC

Chapter 6

APPLICATIONS OF THE DEFINITE INTEGRAL IN GEOMETRY, SCIENCE, AND ENGINEERING

Section 1

Area Between Two Curves

Integrals

Integration

Applications of Integration

Area Between Curves

Volume

Arc Length and Surface Area

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Oh, um, So the important part of this problem is that, you know, the squared of X looks, something like this, and then ah, X squared is the parappa. Look, some doing this, um, this is X squared Ah. Which is important to distinguish which one is the upper function. But then they also tell you that you they only want the area between X equals 1/4 which is just a vertical line to the right and actually goes. One is actually where they intersect each other. Um, so this 1/4 by the way, X equals is a vertical line. So that's why your own vertical. So basically all you need to do if you're finding the area between, um between all of that is ah the integral from the lower round of 1/4 to the upper bound of one. And you do the upper function, which is square to the X right here is above the other and then minus X squared is the lower function right here. We're finding this area. That's all you have to do. Um, now, obviously a little bit more work in this, because the anti derivative of this you add one to your ex bone of one half. That's what the square root is. So that's three halves. And then dividing by that new number is the same as multiplying by the reciprocal of two thirds. Um, and then same thing with this one. You add one to your exponents and you divide by your new exponents and you get one third execute. And that's from 1/4 to 1. Um, so this is where the math gets a little bit more daunting because you have to plug in one for each these exponents. But what's nice is one toe. Any exponent is just one wonder. 30 still one, and then you have to subtract off. Now, this is where it gets a little bit more complicated. Um, two thirds times 1/4 to the three halves. Power my ass. One third times 1/4 the same being cute. I guess I should. Footprint. This is here as well, anyway. Um, so this is easy. Two thirds minus one third is just one third and then this a little bit more difficult, because this means the square root. Um, with the square one. Any power still one. But the square to forest two cubed is 82 times, two times two, and then this woman I don't know if you even know. Maybe one third, um, Times four tens, four tens for 64. Now, I'm not that great moving forward from here. So I'm gonna use my calculator to figure out what, two thirds times 1/8 minus one third time's one 64th ISS. And as a fraction that tells me that that is 5 60 force and one third minus of that answer as a fraction is 4990 seconds. Well, looked weird, but anyway, that's your answer.

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