Problem 46 - Calculus: Early Transcendentals

Question

(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.

Step-by-Step Solution

Answer

Calculate the volume using CAS with rotation about x-axis and the given coefficients.

Step 1: Setting Up the Volume Formula

To find the volume of the egg, which is formed by rotating the curve around the x-axis, we use the formula for the volume of revolution. The volume formula for rotating about the x-axis is given by \[ V = \pi \int_{a}^{b} [f(x)]^2 \, dx \] where \( f(x) = (ax^3 + bx^2 + cx + d) \sqrt{1 - x^2} \).

Step 2: Substitute the Function into the Volume Formula

Substitute the function \( f(x) = (ax^3 + bx^2 + cx + d) \sqrt{1 - x^2} \) into the volume formula: \[ V = \pi \int_{-1}^{1} \left[(ax^3 + bx^2 + cx + d) \sqrt{1 - x^2}\right]^2 \, dx \] where the limits of integration are from \(-1\) to \(1\) because the domain for the square root function \( \sqrt{1 - x^2} \) is \(-1 \leq x \leq 1\).

Step 3: Simplify the Integral

Simplifying the integral involves squaring the function inside the integral: \[ V = \pi \int_{-1}^{1} (a^2x^6 + 2abx^5 + 2acx^4 + (2ad + b^2)x^3 + 2bcx^2 + 2bdx + (c^2 + d^2)) (1 - x^2) \, dx \].

Step 4: Compute the Integral Using CAS

Use a Computer Algebra System (CAS) to evaluate this complex integral from \(-1\) to \(1\). For specific values, when \(a = -0.06\), \(b = 0.04\), \(c = 0.1\), and \(d = 0.54\), input these into the CAS to get the numerical volume of the region.

Step 5: Graphing the Function for the Red-throated Loon

Graph the function using the values \(a = -0.06\), \(b = 0.04\), \(c = 0.1\), and \(d = 0.54\). Ensure the graph is plotted in the interval \(-1 \leq x \leq 1\) and observe the shape of the curve that depicts the egg profile.

Step 6: Calculate the Volume for the Red-throated Loon Egg

Use the CAS tool again to perform the definite integral with substituted values \(a = -0.06\), \(b = 0.04\), \(c = 0.1\), and \(d = 0.54\) into the volume formula. This yields the volume of the red-throated loon egg as a numerical value.

Key Concepts

Definite Integral

A definite integral is a fundamental concept in calculus. It is used to calculate the total accumulation of quantities, like area under a curve or the volume of solid shapes. The definite integral is represented as: \[ \int_{a}^{b} f(x) \, dx \] where

• \(a\) and \(b\) are the limits of integration, showing where the accumulation starts and ends on the x-axis.
• \(f(x)\) is the function whose area under the curve is being calculated.

In the context of finding the volume of a revolution, the definite integral helps in summing up infinitely many small slices of volume, forming the solid shape around the axis. For this exercise, we apply the definite integral to the function describing the egg shape, allowing us to compute its volume accurately.

Graphing Functions

Graphing is an essential tool in understanding the behavior of functions. By visualizing a function, students can see patterns, ranges, and specific values easily, enhancing comprehension. In our exercise, the function given for the egg shape is \( f(x) = (ax^3 + bx^2 + cx + d) \sqrt{1 - x^2} \). To graph this function:

• Substitute the coefficients \(a\), \(b\), \(c\), and \(d\) with their specific values for the red-throated loon.
• Plot the function in the interval \(-1 \leq x \leq 1\), as this is the domain where the square root function \(\sqrt{1 - x^2}\) is defined.
• Notice the curve that forms the egg's profile when rotated around the x-axis.

Graphing the function helps students understand how modifications in the coefficients affect the shape. It's a visual approach to mathematical modeling that brings the abstract function into an understandable form.

Calculus Problem Solving

Calculus is a powerful mathematical tool for solving complex problems involving change and accumulation. The exercise involves several calculus strategies that are crucial in solving the problem of finding the volume of the egg:

• Setting up the mathematical model by representing the egg shape with a function.
• Using the method of solids of revolution to turn the function into a 3D volume problem.
• Applying the definite integral to sum up the components of the volume through integration.
• Using technology, such as a Computer Algebra System (CAS), to handle intricate calculations that are too complicated by hand.

Problem solving in calculus involves not just applying formulas but understanding the underlying concepts so you can adapt these strategies to various types of functions and scenarios.

Mathematical Modeling

Mathematical modeling is the process of representing real-world scenarios with mathematical expressions and equations. In this exercise, we model the shape of the red-throated loon’s egg using the function \(f(x)\). This kind of modeling is useful because:

• It connects abstract math to tangible, natural phenomena, helping to visualize and solve real problems.
• It involves translating physical dimensions and shapes into mathematical terms.
• The model is tested by computing the volume, ensuring it aligns with real-world measurements and expectations.

Through mathematical modeling, complex physical shapes like bird eggs can be studied and quantified, thus providing insights into biological forms by using precise mathematical methods.