5

Find the total area of the shaded region....

Question

Find the total area of the shaded region.

Find the total area of the shaded region.



Answers

Find the area of the shaded region.

Okay for this question we see everything is represented by why, like why X equals y squared minus to an axe. Eco's too either the power Why? So it will be more convenient for us Take the inch grow in with respect to buy So are sheeted region area A very close to the end of our world is with respect Why in the band and go goes from ninety of war here who want and ah, here by the formula is just the upper curve here Ah, there Choose sides to the power by minus the Laura curve the proble life square minus two Okay, so our next step we find ah, anti derivative of that. This will be to Apollo. Why mine is no work. Three like you. Close bye then The value of the pound boundary whyyou goes to war Kanan Lycos Nike Air War. So the result will be Mmm, minus one third plus two Miners need to empower ninety one minus Ah, my as one third times. Like you one Q, which is not a good one. Minus two. Right, So to reach out, uh, there will be cases. Plus here. Those two combined together will give us a four. And this will be too next to third. So minus. But what A To the power ninety one on minus two. Third plus whore quietness to third close four. And we can combine those two together. This's, uh, plus ten, ten third. Okay, me minus, But over me. Close third. Right, So this will be our answer.

Okay. As we see, this problem is represented all the function representing back why on So when they tried to evaluate the interviewer off the shaded region of Integral in terms ofthe why instead of X, um so a region a shader region, he coast too integral in terms ofthe why y in the bound for y ears go. It goes from zero to three. Here is the three here He said origin zero, they go. Well, it goes from zero to three. Ah, and angel voices just up her curved minus the lower curved upper curve. Here is this section which is to y minus y square. So why minus y square? This is the upper curve minus the lower curve is Ritchie's. Why square minus for Why? Okay, so why square liners for what? Um and I ve Kamp Buddha by square together and the y terms together. So this will be zero to three on the double. Minus will be. Plus, So it's six. Why? Minus to y square. You are. And the anti directive for this is, um, three y square, minus two third like you. You ready? And that had the bound Richie Savaiko two three minus like Oh two zero! Ah, and we know when while you go to zero, this whole thing is a zero. So this whole Yuko's Tio with Plug in by three, this will be three times. Ni reaches twenty seven Linus to third times three Q b is twenty seven. So twenty seven. So the answer will be on to third times twenty seven. This is eighteen. Oh, and there will be nice.

Okay. So we can be right. The shaded region as so So the integral and left simplifies down to you 2/3 times X plus two to the power of three. Over to Victor. Evaluate this from zero to minus ln of the absolute value of X plus one on you to evaluate this from zero to a swell. So doing the left hand side. First we end up with 2/3 times four to the power off. Three over to which is this small? Spite by eight minus. Ln of three minus three Star final answer is 16 over three minus ln of three.

Problem. 41. We're given this graph and we're us to find the area of the shaded region. So just recall that the inter grow The definition of an inter go means you're finding the area under the curve. So this kid's were basically asked to find the integral so we can go ahead and right and in a girl so we can see that the bounds are from 0 to 3 and it's the function is given to us as well or minus X squared. D'oh! So this is what we need to find. Then we can do this by looking at the, um graph so we can see that they're two parts one that goes above the X axis and one that goes below. So we're gonna split the inter go the same way, so 0 to 2 is above the X axis. It is your duty to of the function, and then we have one part that's below the X axis, and that's from 2 to 3. So all we have to do is take the integral, so we're going to do that in one step. So the integral of four. There's just four x in the Inter Go of X squared is X Cubed Cube over three and you would evaluate this from the bounds 0 to 2 then plus, the inter go would be the same. However, the bounds would be different. So we have the same integral but the bounds. We're different since it's the area under the X axis. So now all you have to do is plug in the bounds into X. So we have four times two minus 1/3 times, two to the third power minus plug in the lower bound before time zero minus 1/3 times zero cube. And then you add it with the second part you do the same thing four times the upper bound, minus 1/3 three the third power Linus, the lower bound. And if you solve this out, you get 13 minus 16 over three, which gives us 23 over three


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