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Solve the given problems by integration.The perimeter of the rudder of a boat can be described as the region in the fourth quadrant bounded by $y=-0.5 x^{2} \sqrt{4-x^{2}}$ and the $x$ -axis. Find the area of one side of the rudder.

Calculus 1 / AB

Chapter 28

Methods of Integration

Section 8

Integration by Trigonometric Substitution

Integrals

Harvey Mudd College

Baylor University

University of Nottingham

Boston College

Lectures

03:09

In mathematics, precalculu…

31:55

In mathematics, a function…

04:13

For the following exercise…

02:35

Find the area of the regio…

02:29

03:21

07:22

05:02

06:19

Geometry problems Use a ta…

02:53

03:17

01:17

from the graph, I can see that co signed squared is greater than or equal to sine squared. OK, in this particular intell. Okay, once, Once we have this one, then what is the area given by area is It's just in agro cane, the integral just the integral between the two, uh, intervals. Okay, so this one is negative by before and then tired before right here. Okay. And then we have ah co sign since come scientists The biggest biggest a bigger one consign and signed. Okay, because of this inequality here and in the X. Okay, So once you have this one, now we have to find a way to integrate these two. Okay, so we know that CO sign two X is the same as co sine squared X minus sine squared X. Right. We know this one already. So in place of this, you gonna get this one. Okay. Do you know how it became like that? Said that co sign X plus X co sign eggs Coast on exports picks Sign executed guys from this a plus B thing consigned a Who signed me by a sign. A sign? Buk. Yes, I know you got this one. And since we have that one that is going to replace are into grand. Okay, so So place of this one. We're gonna put this one. Okay, so this is gonna be right. This one is easier to integrate. She's gonna be 1/2 signed two X I limit my his lower limit. Right. Okay, so So once we have that one, then what is happening? Uh, this is just gonna be what Have sign. The outer limit is easy when I put probable full here is gonna be 1/2 pie. Okay? Put power before here to Is gonna cancer that take away twice about four. And they just gonna have twice left, then minus lower limit. Some miners over lamented says we're gonna be negative. Okay, I over to, but since sign is a negative, uh, function this negative here is gonna come out and make this one positive. Okay, So why don't you have this on what is signed Piper to say empire, but two is just one and inside pi over two. Here's also one. So this close, this is to So finally you gonna give one right here. Okay, So the area it's actually one. Okay, so that is the area here, straight from here.

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If you write pi (to the first two decimal places of 3.14) backwards, in big, block letters it actually reads "PIE".

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