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(a) A model for the shape of a bird's egg is obtained byrotating about the $x$ -axis the region under the graph of$$f(x)=\left(a x^{3}+b x^{2}+c x+d\right) \sqrt{1-x^{2}}$$Use a 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 volumeof an egg of this bird.

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Calculus 2 / BC

Chapter 7

APPLICATIONS OF INTEGRATION

Section 2

Volumes

Applications of Integration

Missouri State University

Harvey Mudd College

Baylor University

Idaho State University

Lectures

07:20

(a) A model for the shape …

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10:28

03:38

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$\begin{array}{l}{\text { …

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Volume: A model for the sh…

02:07

An egg of = particular bir…

01:31

Volume of a Container A co…

14:28

Bird Eggs The average leng…

09:46

Set $f(x)=\frac{x^{2}}{\sq…

01:47

Volume of a Lab Glass A gl…

04:33

(a) Use a graphing utility…

03:10

A glass container can be m…

02:08

01:53

Write the function for eac…

in part A were given a model for the shape of a bird's egg, and you're asked to use a computer algebra system to find the volume off this egg So the model is obtained by rotating about the X axis the region. Under the graph of the function, ffx, which is a X cubed plus B X squared plus c x Huszti times squared of one minus X squared. Now, to find the volume of such an egg, we want to find first of all, what this region looks like approximately to do this, consider where the region intersects the X axis we know for effect. F of X is equal to zero from the square word term when X is equal to plus or minus one. He's the only zeros that really concerned about. Therefore, the volume Such an egg. Well, the cross sections of this are going to be disks, and so the volume is the integral from X equals negative. One two X equals positive one of pi times The radius of these disks which is their function? A x cubed plus B X squared plus c x plus d times the square root of one minus X squared, All squared DX. This is a pretty complicated into grand. It would take a while to integrate this by hand, but if you use a computer algebra system like Wolfram Alpha, for example, inside pretty quickly, this is equal to four times five a squared plus 18 a C plus three times three B squared, plus 14 times B times D for seven times C squared plus five d squared times pi, and this whole expression is divided by 315. So a complicated but easy to read he's had derivative Part B. We're told that for a certain kind of bird the red throated loon, the values of the parameters, A, B, C and D and were asked to graph the function f and to find the volume of an egg, the species So we're told that is equal to negative. Point of six B is equal 2.4 C is equal to 0.1 and D is equal 2.54 So this is our function. F I was a guest. You use some kind of graphing calculator or computer out of a system to graph this. Otherwise you might miss slight nuances, which I'm going to miss in this sketch. But a rough sketch of this egg looks like this, so I'm only grabbing the top half of the egg. Really? So obviously we have intercepted X equals plus or minus one. We also have an intercept on the Y axis around six. This is really 0.54 We have this. The egg is slightly rounder for positive X, sort of like this and then not as round for negative X. Sort of like that again. This is just a sketch of the curve. I suggest that you use a graphing calculator yourself. Now, if we substitute our values for A, B, C and D into our answer for part a, we get the volume of this egg is going to be 3769 pie over 9375. And this is approximately 1.2 63

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