Problem 72 - Calculus Volume 1

Question

For the following exercises, draw an outline of the solid and find the volume using the slicing method. The base is the region enclosed by \(y=x^{2}\) and \(y=9\) . Slices perpendicular to the \(x\) -axis are right isosceles triangles.

Step-by-Step Solution

Answer

The volume of the solid is 1296 cubic units.

Step 1: Understand the Base Region

The base region is a part of the Cartesian plane bounded by the functions \( y = x^2 \) and \( y = 9 \). This implies solving for \( x \) in terms of \( y \), and for finding intersection points, we set \( x^2 = 9 \), leading to \( x = -3 \) and \( x = 3 \). Therefore, the base region spans from \( x = -3 \) to \( x = 3 \).

Step 2: Sketch the Base Region

Sketch the parabolic curve \( y = x^2 \) and the line \( y = 9 \) on the coordinate plane. The area of interest lies between these two graphs and the vertical lines \( x = -3 \) and \( x = 3 \). This is a bounded region where all slices will be derived.

Step 3: Understanding the Slices

Slices perpendicular to the \( x \)-axis are right isosceles triangles. The legs of each triangle are equal to the distance from the curve \( y = x^2 \) to the line \( y = 9 \), which is \( 9 - x^2 \). The hypotenuse is irrelevant here since it does not define the area.

Step 4: Compute the Area of the Slice

The area \( A(x) \) of each right isosceles triangle with leg length \( 9 - x^2 \) is given by \( A(x) = \frac{1}{2}(9-x^2)^2 \). This comes from the formula for the area of a triangle \( A = \frac{1}{2}(\text{leg})^2 \).

Step 5: Set Up the Integral for Volume

The volume \( V \) of the solid is the integral of the area function over the interval from \( x = -3 \) to \( x = 3 \). The integral expression is:\[V = \int_{-3}^{3} \frac{1}{2} (9-x^2)^2 \, dx\]

Step 6: Simplify and Integrate

Simplify \((9-x^2)^2\) to get \(81 - 18x^2 + x^4\). The integral becomes:\[V = \frac{1}{2} \int_{-3}^{3} (81 - 18x^2 + x^4) \, dx\]Compute the integral term-by-term: \[V = \frac{1}{2} \left[ 81x - 6x^3 + \frac{1}{5}x^5 \right]_{-3}^{3}\]Evaluate the bounds, and simplify to find the volume.

Step 7: Evaluate the Integral

Calculate the definite integral by substituting the bounds:\[V = \frac{1}{2} \left[ \left( 81(3) - 6(3)^3 + \frac{1}{5}(3)^5 \right) - \left( 81(-3) - 6(-3)^3 + \frac{1}{5}(-3)^5 \right) \right]\]Compute and simplify the terms inside the brackets and find the final volume.

Key Concepts

Calculus

Calculus is a branch of mathematics that deals with the study of change and motion. It is divided into two main parts: differential calculus and integral calculus. In the context of finding the volume of solids, we utilize integral calculus. This allows us to sum up small quantities to find cumulative totals—such as the volume of irregular shapes.

• Differential calculus deals with the rate of change and the slopes of curves.
• Integral calculus focuses on summing up small quantities to find totals like area or volume.

When using integrals, you often work with functions and their derivatives to solve real-world problems. Integrals let you calculate the area under curves or the accumulation of quantities over a specific interval.

Isosceles Triangle

An isosceles triangle is a type of triangle with at least two sides that are equal in length. In the context of the slicing method, we use right isosceles triangles, where two sides are equal and one angle is 90 degrees.

• Right isosceles triangle has one 90-degree angle and two equal sides, known as the legs.
• The hypotenuse is the longest side, opposite the right angle, but it is not needed when calculating the area here.

Right isosceles triangles are notable because their area can be simplified using the relation of their legs: \( A = \frac{1}{2} \times \text{leg}^2 \). When solving for volume, this simple formula becomes a key building block.

Volume of Solids

Finding the volume of solids involves calculating the amount of space inside a given 3D shape. Using the slicing method, we can break down complex shapes into more manageable, known forms.

• The slicing method works by taking cross-sections—thin slices—of the solid.
• These slices are shapes we can easily analyze, such as the right isosceles triangle in this exercise.
• These slices build up to approximate the solid, and integrating their areas gives the full volume.

When you work with solids, each slice can vary depending on where it is taken from the solid. For the given exercise, these slices are all the same shape—right isosceles triangles between the parabola and a horizontal line.

Definite Integral

A definite integral is used to calculate the total accumulation of a function over an interval. In volume problems, it sums the areas of cross-sectional slices across the range of interest.

• Definite integrals are written as \( \int_{a}^{b} f(x) \, dx \), representing the area under the curve from \( a \) to \( b \).
• For volume, the function \( f(x) \) could be the area of a slice as derived from isosceles triangle properties.
• Solving a definite integral involves evaluating the antiderivative at the upper and lower bounds.

In the exercise, the integral \( \int_{-3}^{3} \frac{1}{2} (9-x^2)^2 \, dx \) is used to find the total volume by integrating the area of each slice across the defined interval, from \( x = -3 \) to \( x = 3 \). This definite integral brings all the slices together to compute the final volume of the solid.