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Sketch the region enclosed by the given curves. Decide whether to integrate with respect to $ x $ and $ y $. Draw a typical approximating rectangle and label its height and width. Then find the area of the region.

$ y = \sin x $ , $ y = \frac{2x}{\pi} $ , $ x \ge 0 $

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$1-\frac{\pi}{4}$

Calculus 2 / BC

Chapter 6

Applications of Integration

Section 1

Areas Between Curves

Oregon State University

Harvey Mudd College

University of Michigan - Ann Arbor

University of Nottingham

Lectures

05:14

Sketch the region enclosed…

06:16

05:04

02:32

01:25

02:53

09:51

02:31

06:31

Okay. First we observed that we have an e mail here. Machines. Can we only need half? Have playoff the X Y axis. So here is our X y texas. So we're more mainly focus on axis greater or equal to zero. Um, on our first function is signed X. You know, science X goes to pie over too. It goes it attend its maximum and xy coast Opie over too. And the maximum is one. So the science looks like thiss and are lying. Our second function is actually a street line is two x or pie. And we observed that actually close to pie over too. This power to a council with the coefficient. And we got a one richness the intersection off those to function. It's this point. This on maximum, So All right, those op recovery site X and the Lord curvy. Sorry. Straight line and are in close. A region will be the shaded regions here. Um, and do you represent her ex? So we decided to integrate with the respect to ACS. Um, draw typical approximating rectangle jaw, for example. We pick our tango right here, and you three zoom in. Zoom out. Actually, two out look like this. The the wealth will be Delta X. Huh? The mouth is dealt acts. This is some else. And, uh, had the height will be the upper curve, Cy. Thanks. I minus the lower curve. The Lord curries are a straight line two X, I or pie. Okay, Um um then they need to evaluate this area. How do you do that? I know the formula will be a coast to injure role since we decided to integrate with the respect you ass ex put R D X here and the boundary for X. It goes from zero to two. He goes from zero to pie over too. Ah, here. We simply put this this heart without the sub script, which is sai X minus tow x or a pie. Okay, um, when we were flying the anti duty about that, it will be on active coastline acts minus x square over pi. All right, this is our anti derivatives and matter XY coast to pirate too, And exit code zero. So xy coast power to win No co sign. Power to is zero. So this will be just a negative pi square over four. So it will be pi square over Pi square over for over pi missions. This pirate for this is those pilot cancel. So we got pie on the top and two squares for right and minus. We're plugging X equals zero. Um, so minus one coast zero is one in this term skull. So he's just aa minus ninety Roban. And we got minus minus canceled. And we got one minus pi over four, Yeah.

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