Problem

The figure here shows triangle $A O C$ inscribed …

Problem 78 Hard Difficulty

Find the area of the region in the first quadrant bounded on the left by the $y$ -axis, below by the curve $x=2 \sqrt{y},$ above left by the curve $x=(y-1)^{2},$ and above right by the line $x=3-y$
(GRAPH NOT COPY)

Answer

$\frac{5}{2}$

Related Courses

Calculus 1 / AB

Calculus 2 / BC

Thomas Calculus

Chapter 5

Integration

Section 6

Substitution and Area between Curves

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Video Transcript

So, since they give us everything in terms of why they're so X equals action makes sense. Toe, you know, do things in terms of why. So the first inning Greul you can see in terms of why goes from 0 to 1. And that's just the function to square root of why outright is why to the one half power. And then when you do the integral from 1 to 2, because everything's in terms of why the upper function is your furthest to the right. I guess I should say, is the three minus wife function and then subtract off the other function. That's to the left. Which is that why minus one being squared? Do you? Why, Before doing this problem, I would clean up this dysfunction just because I don't really like that. Why minus one being squared. So rewrite and your foiled out b y squared minus two y plus one. But don't forget that you're subtracting off each one of those terms. So as I distribute that in, it's going to become negative y squared and, um, negative y plus two y and then three minus one plus two. Do you know why? And then it's pretty easy now, I think, just to do each problem each piece to my no. One half d y and then add the areas together. So as you add one to your exponents, three. Have divided by a new exponents or multiply by their sickle. Two times two is four thirds and that's from zero. The one and then add to it this which is negative. One third. Why Cube plus one half. Why squared plus two. Why? And that's from 1 to 2. Um, so this is nice. Just plug in. Want any power at one? So you just have four thirds there and then plugging into this is gonna get kind of ugly for me. But to Cuba's eight name eight thirds C two squared is four has is two plus four there minus, um, plugging in. One is going to keep everything the same. Negative one third plus one half plus two. Um, let's hopefully I'm doing this. Right, um, in getting the same denominator, don't forget to distribute this year there. I'm notorious for forgetting, but four minus eight is native four plus one to be negative. Three thirds. So that's negative One. Yes, I took care of that. That and that C two plus four is six minus this to be back to four and then minus that one half. So that be three minus one half is 2.5. But most math teachers would write it as a improper fraction instead of a mixed number. Uh, which matches, I believe. The answer key, I think it says 15 6. Which wonder doing reduce the five halves?

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