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# (a) A model for the shape of the bird's egg is obtained by rotating about the x-axis the region under the graph of $$f(x) = (ax^3 + bx^2 + cx + d) \sqrt{1 - x^2}$$Use $CAS$ to find the volume of such an egg.(b) For a red-throated loon, $a = -0.06$, $b = 0.04$, $c = 0.1$, and $d = 0.54$. Graph $f$ and find the volume of an egg of this species.

## a) $V(x)=\pi\left[\frac{4 a^{2}}{63}+\frac{8 a c}{35}+\frac{4 b^{2}}{35}+\frac{8 b d}{15}+\frac{4 c^{2}}{15}+\frac{4 d^{2}}{3}\right]$b) $V(x) \approx 1.263$

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Applications of Integration

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a model for the shape of the bird's egg is obtained by rotating about the X axis. The region under the graph of F. Of x equal eight times x cubed plus V. X square. The C. A. Ex bestie That value times squares of 1 -6 square. That is F is a product of polynomial degree three Times discovered of one -X square. We're going to use cast to find a volume of such an egg. You are a barbie for a particular species. The red throated loon We have eight equals negative. 0.06 equals 0.4 C equals european. One D equals 0.54. Bad particular values. We graph F and find the volume of an egg of these species. So uh part A. We know that the volume is generated by rotating these. If the graph of this function around or about the X axis first we determine where this function is positive and define well for this function we know that the domain it's uh determined by He is square root of 1 -6 square. Because the first part, the polynomial degree three is to find everywhere independently of the coefficients A, B, C and D. So for the square root of one minus x squared to be well defined in the real numbers, 1 -6 square, going to be no negative. That is greater than or equal to zero. So the main ff is the real numbers such that 1 -6 sq. He's very greater than or equal to sue. That's the same as the real numbers that are Or half square created than less than or equal to one. Okay. And if it takes it would take a square. Both sides of this inequality. We get the real numbers such that the absolute value of X Is less than or equal to one. That's the same as the closed interval from -1 to walk. So the function is defined there. And in that case we know that for some values of parameters we will have a positive graph of the function. So in general we know that the volume of the region in space determined by retaining about its access. This region determined by dysfunction and the X axis. We know this is given by the integral from negative 1 to 1 which is liberation of X. And then we have five times the function itself square because we have the areas of circles something like this. It's a function has for example this shape here. So we will take this way. So um we will add circles determined by or with radius equal the value of the function at each point. So this will be the diameter and we integrate those circles between mhm um limits or definition of the function or the limits we have to find previously. So This case these limits are 91 1 and then we have the areas of the circles are by times the radius square and the way this is the image through the function F of each point X between M and N. In this case between the 81 and more. So that's it. This expression so this is the injury from 81 to 15 times A xQ plus B X square plus E X nasty times square to 116 square. All that square because we have a product here and by the constant we take it out of the integral. We had five times into reform negative 11 of a X cubed plus B X square plus C. X plus D. Square times and discovered one minute six square square is one minus six square. We could do this. I have this tedious because we had four terms in a square and then a multiplication is very tedious to the to the soul. And we use a calculator or algebraic system in the computer. For example of views. Mad lab, you can use maple or another one. Mad lab in fact has maple inside of it. So I'm missing maple. Uh in reality so um using that cool, we have found that this integral is equal to for by over 135 times 58 square plus 18 A C plus nine be square plus 42 B. D plus 21 C square Last 105. The square. That's it. So his expression we get if we do this in general here behan but he's better to use chaos computers generate system and better find written correctly all all the terms. That's it. And now we can do a particular calculation. Once we have this it's easy to calculate the volume for specific species we have. What we have to do is replace each of the coefficients A, B, C and D. Here. This formula we have the song. So for these particular species we did that the volume is equal to 3000 700 69. Bye Over 9300 75 Which is about one 26. three serious 3 or four 0- 2 5. I have given 10 decimals here. So this is the miracle value of the volume of the egg of the particular species given in the statement red throated bloom and the general volume was calculated here is in computer algebra system. And we found this expression here which have been on the coefficients A, B, C and D. And that gives us the volume of the egg resulting from rotating the region determined by the function given function and the X axis. The rotation is about the X axis. Mhm