Problem 12 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

We model a large, 9 -inch-tall soup bowl by rotating a region \(A\) around the \(y\) -axis. \(A\) is the region bounded by the \(y\) -axis, \(y=\frac{1}{4} x^{2}\), and \(y=9\). (a) What is the capacity of the bowl? (b) If the bowl was filled to the brim when set out on the table and an hour later was filled only to a height of 4 inches, how much soup was ladled out in the hour?

Step-by-Step Solution

Answer

The capacity of the bowl is 162π cubic inches while the volume of soup ladled out in the hour is 32π cubic inches.

Step 1: Setup the Integral for Total Volume

The radius of each disc is \(x\) and height is \(dy\). The volume of each disc is \(\pi x^2 dy\), where \(x=\sqrt{4y}\). Therefore, the total volume \(V_1\) of the soup bowl is given by the integral from 0 to 9 of \(\pi x^2 dy = \pi \int_{0}^{9} (4y)\,dy\).

Step 2: Evaluate the Integral for Total Volume

Evaluating the integral gives, \( = \pi \int_{0}^{9} (4y)\,dy = \pi \left[2y^2 \right]_0^9 = \pi [ 2(9)^2 - 0] = 162 \pi\). This means \(V_1 = 162 \pi cubic\,inches\)

Step 3: Setup the Integral for Remaining Volume

The remaining bowl from y=4 to y=9 gives a new volume \(V_2\). Let's setup the integral as in step 1, but this time from 4 to 9, \(\pi \int_{4}^{9} (4y)\,dy\).

Step 4: Evaluate the Integral for Remaining Volume

Evaluating the integral gives, \(\pi \int_{4}^{9} (4y)\,dy = \pi \left[2y^2 \right]_4^9 = \pi [ 2(9)^2 - 2(4)^2] = \pi [162 - 32] = 130\pi\). This means \(V_2 = 130\pi cubic\,inches\).

Step 5: Determine the Volume of Soup Ladled Out

The amount of soup ladled out is \(V_1 - V_2\). Therefore, the amount of soup ladled out in the hour is \(162 \pi - 130 \pi = 32 \pi cubic\,inches\).

Key Concepts

Volume of Revolution

When we talk about the **Volume of Revolution**, we are referring to a fascinating aspect of geometry that involves creating a three-dimensional figure by spinning a two-dimensional shape around a line. In the exercise above, we are asked to consider a region in the coordinate plane and rotate it about the y-axis to form a bowl shape. This idea can be thought of like the act of potters making a vase by spinning clay.

Here’s how it works: You take a curve, such as our given parabola, and use the rotation to form a symmetrical object. The line (or axis) of revolution acts like a spine around which the shape is mirrored. For calculation, the volume of this revolved shape becomes an integration problem that requires understanding how slices of the object (discs in this case) add up to form the entire volume.

The underlying goal is to use calculus to determine the space that this three-dimensional object occupies, which is particularly useful in physical sciences and engineering.

Definite Integrals

**Definite Integrals** play a pivotal role in calculating the volume of revolution. In essence, a definite integral helps us to determine the total value of a function over a specific interval. In the context of finding volumes, this is about summing up an infinite number of infinitesimally thin slices or discs. These integrals provide a means to account for the gradual change in volume as we move along the axis of rotation.

In simpler terms, think of a definite integral as a sophisticated way of adding up all the tiny bits of volume (or areas) that are found under a curve between two points on an axis. For the soup bowl problem, this involves taking the function that describes the curve, manipulating it as necessary to fit the volume formula, and then evaluating the integral over a specified range, such as 0 to 9 for the entire bowl. Tools like the Fundamental Theorem of Calculus make these calculations feasible.

• It determines total accumulation, such as area or volume.
• The limits of integration signify start and end points of the calculation.
• Helps transform geometric problems into arithmetic ones.

Disc Method

The **Disc Method** is a technique that simplifies how we calculate the volume of a solid of revolution. Imagine slicing the solid into many thin discs (like coins) along the axis of rotation. By calculating the volumes of these individual discs and then summing them all, you can find the entire volume of the object.

In the given problem, the radius of each disc is determined by the distance from the y-axis to the curve of the parabola. As you might have guessed, this varies along different points of the axis, which is why integration is necessary. Mathematically, the volume of each disc is given by \( \pi x^2 dy \), where \( x \) is the variable radius and \( dy \) is the infinitesimal thickness of each disc.

• It's useful for solids that's symmetrical around an axis.
• Works well when slicing perpendicular to the axis of revolution.
• Relies on knowledge of functions and their inverses for radius determination.

Applied Calculus

**Applied Calculus** is where theory meets the real world, solving practical problems using calculus methods. From calculating soup bowl capacities to designing engineering projects, applied calculus helps in modeling real-life situations with mathematical clarity.

In our soup bowl scenario, applied calculus allows us to not just understand the abstract concept of rotating a region, but to pin down exactly how much space is inside that bowl. If you wanted to fill this bowl with liquid, how much would it hold? And what if some of it is removed? This requires stepping beyond pure theory into the realm of everyday problem-solving where calculus becomes a tool rather than just a study.

• Helps in creating models that describe real-world phenomena.
• Makes abstract mathematical concepts applicable to physics, engineering, economics, etc.
• Involves predicting outcomes and understanding trends using mathematical models.