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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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### Video Transcript

Okay, We know that via fax can be written as pie times The integral from negative 1 to 1 of X cubed was B X squared asiax pastilles, times square root of one minus X squared DX, which means we're integrating from negative 1 to 1. Therefore, we know we can also write this using C A s as pi times for a squared over 63. A c over 35 us for B squared over 35 was a P D over 15. Now, looking at part B, we already know what we have for part A. Therefore, we can simply substitute given values for A, B, C and D. And then if you want a graphical visual, you have this. Which means you can also said you can use a graphical visual as well to figure out via axe. If this is easier for you can put this into a formula like into a program like gizmos, for example, and you should get 1.263 or you can plug in directly A, B, C and D

#### Topics

Applications of Integration

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