Problem 20 - Basic Technical Mathematics with Calculus

Question

Find the volume generated by revolving the regions bounded by the given curves about the \(y\) -axis. Use the indicated method in each case. $$y^{2}=x, y=5, x=0 \quad(\text { disks })$$

Step-by-Step Solution

Answer

The volume is \( 625\\pi \).

Step 1: Identify the Region

The curves given are \( y^2 = x \), \( y = 5 \), and \( x = 0 \). The region bounded by these curves in the first quadrant is under consideration: the parabola \( y^2 = x \), the horizontal line \( y = 5 \), and the vertical line \( x = 0 \).

Step 2: Set Up the Integral

Using the disk method about the \( y \)-axis, the volume \( V \) is given by the integral: \[ V = \pi \int_{a}^{b} [R(y)]^2 \, dy, \] where \( R(y) \) is the radius of the disk. Here, from \( y^2 = x \), the radius \( R(y) = y^2 \). The given bounds for \( y \) are \( y = 0 \) to \( y = 5 \).

Step 3: Evaluate the Integral

Substitute \( R(y) = y^2 \) into the volume formula: \[ V = \pi \int_{0}^{5} (y^2)^2 \, dy = \pi \int_{0}^{5} y^4 \, dy. \] Integrating, we have \[ V = \pi \igg[ \frac{y^5}{5} \igg]_{0}^{5}. \]

Step 4: Compute the Result

Now calculate the result: \[ V = \pi \igg[ \frac{5^5}{5} - \frac{0^5}{5} \igg] = \pi \igg[ \frac{3125}{5} \igg] = \pi \ imes 625. \] So, the volume generated by the revolution is \( 625\pi \).

Key Concepts

Disk Method

The disk method is a technique used in integral calculus to find the volume of a solid of revolution, often when a region in the plane is revolved about an axis, converting it into a 3D object. The method gets its name because the solid is thought of as consisting of disks stacked along the axis of rotation.

Here's how it works:

• You identify the function that defines the boundary of the region you're rotating. In many cases, this will involve defining the radius of these disks. The radius at any point will be the function value at that point.
• You set up the volume integral. For rotation around the y-axis, disk radius is a function of y, often noted as \( R(y) \). The formula for the volume \( V \) when using the disks method around the y-axis is: \[ V = \pi \int_{a}^{b} [R(y)]^2 \, dy \]This integral essentially adds up the volumes of an infinite number of infinitely thin disks stacked from \( y = a \) to \( y = b \).

Using this method helps simplify the calculation by breaking down the 3-dimensional problem into a manageable series of one-dimensional calculations.

Volume by Rotation

Calculating volume by rotation is a powerful application of calculus that allows us to determine how much space an object occupies when a planar region is rotated around an axis. This geometric transformation turns a 2D area into a 3D solid.

To set up your problem:

• Identify the shape of the region being rotated, which often relies on its boundary curves, such as a parabola or a line.
• Determine the axis of rotation. This could be the x-axis, y-axis, or any parallel line. The axis choice dictates the method: disks, washers, or shells.
• Choose the correct formula and setup for calculating the volume. Each method has specific forms - for the disk method, the volume of each disk is connected to its radius and thickness.
  Once identified, you integrate over the interval you are rotating, accounting for the entire solid.

Volume by rotation requires a keen understanding of setting up integrals correctly since the axis influences the function limits.

Integral Calculus

Integral calculus is a subfield of calculus concerned with the concept of integration. While different from its counterpart, differential calculus, integration often involves finding the whole from its parts, such as calculating the total area under a curve from many infinitesimal slices.

Important aspects include:

• Definite and Indefinite Integrals: A definite integral computes a net quantity over an interval, such as area under a curve, typically written as \( \int_{a}^{b} f(x) \, dx \), whereas an indefinite integral finds an antiderivative without specific bounds, \( \int f(x) \, dx \).
• Fundamental Theorem of Calculus: This connects derivatives to integrals, stating that differentiation and integration are inverse processes. It provides a way to evaluate definite integrals using antiderivatives.
• Riemann Sums: This is the foundational approach for understanding definite integrals, approximating total values based on a sum of individual parts. The more partitions, the closer the approximation to the true integral.

Integral calculus is essential in calculating volumes by rotation, as it allows us to sum the continuous distribution of infinitesimally small volumes to get the solid's total volume.