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J zdz where € is the semicircle fromi,which passes through...

Question

J zdz where € is the semicircle fromi,which passes through

J zdz where € is the semicircle from i,which passes through



Answers

Locate the centroid $\bar{y}$ for the beam's cross-sectional area.

Question 20 would like you to evaluate the double integral of X Y d A. Where d is enclosed by the quarter circle y equals square root one minus x squared where X is greater than or equal to zero and the axes So drawing that on our graph here this will be 11 and this is why equals group one minus X squared And then the axes you have y equals zero X equals zero. This is our region here, um, writing this as a type one and a girl which is the easiest to solve in the X direction. You're going from 0 to 1 and in the Y direction you're going from zero to square root one minus X squared x y dy dx From there taking that first integral so 01 of X Y squared over two from zero to square root one minus x squared d x plugging in. You have 0 to 1 x times one minus x squared. We're two d x, which is equivalent to any girl, 0 to 1 of X over two minus X cubed over to DX Now taking your second in a role you have X squared over four minus X to the fourth over eight from 0 to 1, just just 1/4 minus 1/8 which is equal to what age and that is your final answer to question 20.

For problems in center of gravity and central, you have a composite body that is an angle and were given its cross section. The wind off the angle is a with a thickness off the So let us have divide this cross section in tow three parts. So we have this square part and on identical legs It's called the legs ass segment one nd square ass segment toe. So we will tie. Believe the area and there corresponding zand trades. So we have a segment one which are identical and segment toe which is a square. Their area is the area off. The square is D squared in the area off the rectangle is a minus t multiplied by the shall we have a D minus the square a T minus d squared in their corresponding century. Eso let's blow up the cross section So we have this society and the society nd centered off the square. So let's call this distance is why too so therefore, this is also why, too, using Pythagorean theorem, we have right to squared brush by two square this equals d squared. So solving for a way to we would then have the square it off to all over to multiplied by the thickness and for the legs. Let's extend this drawings so that you can include the segment one. So we have this SD center it off Segment one. So let's recall that we can get this distance. Why? Which we will call us. Why one and we can also get this distance from the central AIDS and we know that this angle ish 45 degrees. So first, let's determine the hype Artiness. We know that this distance is equals toe a minus t Well, I'm very cool. Therefore, the other side is also a minus. T what, over two. And this distances t all over tow. So we have now the I part in us which is it was too in my honesty, all over to us d all over to giving us with only a all over toe. So now we can determine why one which is supposed to sign 45 multiplied by a 1/2, giving us a squared off to well over four times e. But this isn't Lee sent tradeoff the rectangle we need to add y to tow it. This is why to we need to add vital to it. So determined the sen trade off the segment off the legs. We need to add by one end. Why to So we have square it off to all over to times de los square it off to all over two times a day so we can factor Alex square it off to on over four. So we will get a plus two times d. So they have squared off to over four times a minor plus booty its creditors school labor for a plush 30. And for a segment toe, you have square it off to Loreto times t You can now multiply the area to Yeah, Why bar so we wouldn't get discredit off to a lumber for was played by a squared D plus to a T squared minus a D squared minus two d cube. And we can simplify this so we can subtract this, too. And then we can father out the so we would have square it off to a lumber for times D multiplied by e squared plus 80 minus two de screwed. So we have square it off to lumber for thanks D multiplied by eastward plus 80 minus toe the squared. And this one is square it off to all over to thank you. So now we can do the submission off area. So bad. The summation off area we have to 80 minus to d squared plus three squared so we can subtract this. So we have to 80. I know Dusty squared for the summation. Off a way bar. We have the square it off to all over to times D multiplied by a squared plus 80 minus to d squared, plus the square it off to whatever Toe Cube. So by doing the same mention off a way bar divided by the submission off A we wouldn't get the y bar. So we have our values as squared off to all over to so we can fucker at squared off the all over two nd multiplied by a square glass 80. I understood T Square lost t scream. We can simplify this. We didn't have negative the squared divided by to a D minus described, so we can see that you can simplify this more so simplifying it would give us the final answer, which is squared off to a lover too. multiplied by a square plus 80. My honesty squared, divided by to a dynasty. So this is the final answer.

Okay. Were asked to find the century of a circle or a semicircle with radius are its center of origin. Jake misses. Things are all right. I have half a circle. Okay, So since this is a semicircle with center at its origin, we know that a question for a circle at its origin is X squared. Plus y squared is you two are square. Um, since if we look at the y axis, this portion in this portion is the same. We know that based on cemetery about the wire. Since there is a metric we know that our ex CM is equal to now let's look for area. That area for a circle is equal to high r squared. But since we have half a circle, that's over too. So half a circle and you cook it pi r squared over two. But if you got em, you know, looking for So we have our exam and our area. We just need to look for Emelec's That's equal to in a girl. Negative arch are 1/2 p. We're using a question, Jerry. It's all for this in terms of wine. So we get r squared when it's X squared square boot You go to what? So that's equal y eyes open. Plug this in. That's R squared minus X squared her to the ex. Okay, that gives us 1/2 p. You know, go from negative arts are of r squared when it's X squared. T ex Excuse me. R squared Epps finished Execute over three evaluated at or in negative or all times? 1/2 p. Oh, you should have just assumed that PC puts one in this case, cause and nd still cancel assuming that he is one with Dr P. So we have one Have our origins are squared, isn't r cubed minus r cubed over three warning US evaluation that negative are which is r squared or negative are cute minus, plus or cubed over three Combining these That's two r cubed over three minus plus two r cubed over three That gives me for our cue over three times 1/2. Excuse me to R cubed over three. This is Emma Vex. So you know M is equal to for pie. You believe? Oh, no pirates great over too And X is equal to two are accused open three and we know that ex of em Xia Medical Journal based on cemetery. So exhale comma y como tickled to r cubed over three divided by m which is two over hi r squared. Okay, our powers of are canceled. We left with offto zero comma Your comma for our older three, huh? This is the center of mass for a semicircle with reduced our sense it at the origin.

Okay, We know that if we're looking at a squared plus B squared equals C squared, we can apply the same therm in this case, 17 squared R right triangle. Ph Cube equals eight squared plus que h squared. Right? So then we know that cute h is gonna be 15. Remember? Qh and Jacare equivalent. Therefore, the answer is 15.


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