#### To determine

**Part (a):**

**To find:**

The volume of the shape of a bird’s egg by rotating about the x axis the region under the graph of fx=ax3+bx2+cx+d1-x2 by using a CAS

#### Answer

V=4π5a2+18ac+9b2+42bd+21c2+105d2315

#### Explanation

**1) Concept:**

Use definition of Volume

**2) Definition of volume:**

Let S be a solid that lies between x=a and x=b. If the cross sectional area of S in the plane Px, through x and perpendicular to the x-axis, is A(x), where A is a continuous function, then the volume of S is

V=limn→∞∑i=1nAxi*∆x=∫abAxdx

If the cross section is a disc and the radius of the disc is in terms of x or y then area

A(x)=π radius2

**3) Given:**

fx=ax3+bx2+cx+d1-x2; about the x axis

**4) Calculation:**

Set fx=0 to find limits of integration since graph meets x axis when fx will be zero and shape of a bird’s egg lies between those x values.

ax3+bx2+cx+d1-x2=0

1-x2=0

Square both sides of the equation and solve for x

1-x2 =0

x=-1 and x= 1

The solid lies between x=-1 and x= 1

By using the concept, the volume of the shape of a bird’s egg is

V=π∫-11ax3+bx2+cx+d1-x22dx

By using the command in Mathematica,

Integrate[Pi*((a*x^3+b*x^2+c*x+d)*(Sqrt[1-x^2]))^2,{x,-1,1}]

=4a2π63+4b2π35+8acπ35+4c2π15+8bdπ15+4d2π3

=π4a263+8ac35+4b235+8bd15+4c215+4d23

=4πa263+2ac35+b235+2bd15+c215+d23

V=4π5a2+18ac+9b2+42bd+21c2+105d2315

Therefore,

The volume of the shape of a bird’s egg is

V=4π5a2+18ac+9b2+42bd+21c2+105d2315

**Conclusion:**

The volume of the shape of a bird’s egg is

V=4π5a2+18ac+9b2+42bd+21c2+105d2315

#### To determine

**Part (b):**

**To graph:**

f by using a=-0.06, b=0.04, c=0.1, and d=0.54 then findthe volume of an egg.

#### Answer

Graph of fx=(-0.06x3+0.04x2+0.1x+0.54)1-x2):

V≈1.263

#### Explanation

**1) Concept:**

i. If the cross section is a disc and the radius of the disc is in terms of x or y, then area A=π radius2

ii. The volume of the solid revolution about the x-axis is V= ∫abA(x)dx

**2) Given:**

fx=ax3+bx2+cx+d1-x2 ; about the x axis

and a=-0.06, b=0.04, c=0.1, d=0.54

**3) Calculation:**

Substitute a=-0.06, b=0.04, c=0.1, and d=0.54 in fx

fx=ax3+bx2+cx+d1-x2

fx=(-0.06x3+0.04x2+0.1x+0.54)1-x2)

Graph of fx=(-0.06x3+0.04x2+0.1x+0.54)1-x2)

By using part (a),

V=4π5a2+18ac+9b2+42bd+21c2+105d2315

Substitute a=-0.06, b=0.04, c=0.1, d=0.54 in above step.

=4π5-0.062+18-0.060.1+90.042+420.040.54+210.12+1050.542315

Solving this V≈1.263

Therefore,

The volume of an egg for a red throated loon is V ≈1.263

**Conclusion:**

The volume of an egg for a red throated loon is V ≈1.263