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Question 15volume described as follows: the base the region bounded by = v + 14y 33and < = % z0y 107; every cross section perpendicular t0 they-axis is a semi-ci...

Question

Question 15volume described as follows: the base the region bounded by = v + 14y 33and < = % z0y 107; every cross section perpendicular t0 they-axis is a semi-circleFind the volume of this objectvolumeQucstion 16The graph above shows the base ol an object Compute the #ax ulua of the volume of the obf-c given that cross sections (perpendicular t0 the base) are equiangular trianglesQuestion 19Find the volume of the solid obtained by rotating the region in the first quadrant bounded by the curve

Question 15 volume described as follows: the base the region bounded by = v + 14y 33and < = % z0y 107; every cross section perpendicular t0 they-axis is a semi-circle Find the volume of this object volume Qucstion 16 The graph above shows the base ol an object Compute the #ax ulua of the volume of the obf-c given that cross sections (perpendicular t0 the base) are equiangular triangles Question 19 Find the volume of the solid obtained by rotating the region in the first quadrant bounded by the curves I =09=1= $ about the line %



Answers

Find the volume of the solid whose base is the region in the first quadrant bounded by $y=4-x^{2},$ the $x$ -axis, and the $y$ -axis, and whose cross-section in the given direction is an equilateral triangle. Include a sketch of the region and show how to find the area of a triangular crosssection. Perpendicular to the $y$ -axis.

The origin bounded by. Why call to ex choir. Why call to one and Y. Axis four X. Greater than an equal to zero. So here we have to find the volume of the solid and the solid whose base is the origin and whose cross section perpendicular to the X axis are same circle. So here you can see that from the growth all equal to one negative ex choir upon. Do so here you can see that given X. Xs are semi circles so volume of slice approximate E. Time delta X. And here we know that equal to why time ari Squire upon to So here equal to by the time our Squire upon to time delta X. Now we put the value of our So here we get five upon two times one negative X. Choir upon to to the whole power to time. There's the X. Those volume of solid approximate value sigma by a 0.2 dying one negative X. Choir to the power to upon four time delta X so here there you go. X thickness off each slice tends to zero. The sum becomes a definite intrigue. Als since curves intersect ed X equals do negative one and X equal to one So here volume of solid equal to we so we put here we equal to definite intrigue. All of by upon eight time one negative two times X choir plus active Squire. So we actually about four dx from negative 1 to 1. So now we have to find that everybody of one negative two times X squared plus actually about four is equal to 5.8 time X negative two upon three time XQ plus. Actually about five upon five where negative one 21 so you can see that upper value is one and lower value is negative one. So first of all report upper value. So here we get weak will do by upon a time one negative two upon three positive one upon five dime negative 5.8. Now here lower value is negative one. So we put here X equal to negative one. So here we get negative one negative negative positive two upon three negative one of on five. So we get equal to to buy a bond 15. That is the value of solid, so it is over final answer.

So we have, he expects, is and we have the y axes and then on the vaccines be at this curve, which is why is he could through one minus squared Ruto fix one minus x quay under the square. So the places where it across the X axis would be when X is equal One hand excessive, well connected one. So on this we take a line which is indicator to be X axis. And on that line we will square and the length of this square will be quitters. We're going one minus X squared and the thickness off. This is D X. So we start by first of oil saying that l is equal to one minus x squared under the square root and the area would be the square off that'll l squared at Syria So one minus X squared and the square root squared gives me one like this X squid and the wild is equal to a X times DX or D V is equal to one minus x squid ni Exe So now, in order to find the volume, we just need to apply the limits from negative one to one in a grilled one minus X squid dx. So why would be equal to X minus X to the power three over. Really over. Be limits from negative one tube positive one. So V is equal to one minus one Orphan E minus negative. One minus negative. One. Cute over three. And that's he could one minus 1/3 minus negative. One minus up, plus Waterworth. So that is equal to one minus 1/3 for this one minus one. Or this is a two to minus 2/3, and that's equal to for.

We have a region region bounded by Why I called it Swede and line We have a lying right to be able to when right to the equal to one we want to find a value. Oh, it's so early Only I seen that the sections have been deployed to the bees and parallel to the eggs and sealed with Lola Indeed sued day bees. And it's no so Parral. So day ace words chances. So this is a way Have so saints saves One is he said, that oh goods and Richwoods worry. That's beautiful, Zo, Why should we exert a little place than one? This is our words our insatiable you can done writes our folio G I wouldn't be here to be called so by they're excellent for Dane said Well, we have way. If if was worry the way which is records. What is every boy f off? Why here By good 01 beautiful boy. It was why screed, right? It's great! The way just okay, right? You the way. Let's find a in sever being situation. What do you have? You have Why incident for we're full from what zero to woods or if you simplify? Do I have? It was The one is buried up because deserves I have by one over words just people I full so base they is the folio. We're looking full civilian off a solid.

Okay, So for this problem, we're looking at the equation. Why equals X Square is our X to the third and then between Why equals zero and why equals one? So what we want to do is we're gonna look at the the graph of y equals X to the third, and we're looking at it between why equals one and then, of course, the Axis. But we're looking at this in terms of the the Y Axis. So we're looking at this little piece here. And so what we want to do is we wanna look at the volume, so we're going to the integral between zero and one. Now, since we're doing this about the why Axis, that means I want to look at the inverse of y equals X to the third. So that's going to be X equals y to the one third or the cube root. So this is going to be why did the one third squared de y? So now, of course, this is going to end up being the integral from 0 to 1 of why to the two thirds power do y. And so this is going to be why to the five thirds because I'm adding three thirds to it. And then what I also want to do is I want to multiply this by the reciprocal of the exponents, which will be 3/5. And I'm looking, of course, of this between zero and one. So obviously, um, once of the five thirds power is going to be one and then zero to the five thirds power of B zero, so the answer would be


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