Question

Find the volume of the solid generated by revolving about the \(x\) -axis the region bounded by the line \(x-2 y=0\) and the parabola \(y^{2}=4 x\)

Short answer

The volume of the solid is approximately \(107.18\).

Step-by-step solution

Solve for Intersection Points

To find where the line and the parabola intersect, we substitute the equation of the line \(x = 2y\) into the equation of the parabola \(y^2 = 4x\). This gives \(y^2 = 4(2y)\). Simplifying, we obtain \(y^2 - 8y = 0\), which factors to \(y(y - 8) = 0\). Therefore, \(y = 0\) or \(y = 8\). Then, substituting back to find \(x\), we find the intersection points are \((0, 0)\) and \((16, 8)\).

Set Up the Integral for Volume

The region is revolved around the \(x\)-axis, so we will use the disk method. The formula is \( V = \pi \int_a^b [R(x)]^2 \, dx \), where \(R(x)\) is the distance from the \(x\)-axis to the curve. For each curve, solve for \(y\) in terms of \(x\): the line gives \(y = \frac{x}{2}\) and the parabola gives \(y = \sqrt{4x}\). The radius of the disk is the difference of these two: \(R(x) = \sqrt{4x} - \frac{x}{2} \).

Determine the Limits of Integration

From Step 1, the curves intersect at points \((0, 0)\) and \((16, 8)\). Hence, the limits of integration for \(x\) are from \(0\) to \(16\).

Integrate to Find the Volume

Substitute \(R(x)\) into the integral formula: \[ V = \pi \int_0^{16} \left( \sqrt{4x} - \frac{x}{2} \right)^2 \, dx \]. First, expand the term: \(\left( \sqrt{4x} - \frac{x}{2} \right)^2 = 4x - 2x^{3/2} + \frac{x^2}{4}\). Integrate each term individually: \(\int 4x \, dx = 2x^2\), \(\int -2x^{3/2} \, dx = -\frac{4}{5}x^{5/2}\), and \(\int \frac{x^2}{4} \, dx = \frac{x^3}{12}\). Evaluate this from 0 to 16.

Evaluate the Definite Integral

Substituting the limits of integration, for each component after integration, we get \[ \left[ 2x^2 - \frac{4}{5}x^{5/2} + \frac{x^3}{12} \right]_0^{16} = \left( 2(16)^2 - \frac{4}{5}(16)^{5/2} + \frac{(16)^3}{12} \right) - (0) \]. Calculating these: \(2(16)^2 = 512\), \(\frac{4}{5}(16)^{5/2} = \frac{4}{5}(1024) = 819.2\), and \(\frac{(16)^3}{12} = \frac{4096}{12} \approx 341.33\). Thus, \[ V = \pi ( 512 - 819.2 + 341.33) \].

Compute the Final Volume

Adding the terms in the integral, you find: \(512 - 819.2 + 341.33 = 34.13\). Thus, \( V = 34.13\pi \). So, the volume of the solid is approximately \(107.18\).

Key concepts

Disk Method

The Disk Method is a fundamental technique in integral calculus used to determine the volume of a solid of revolution. Imagine a solid formed by rotating a region about an axis, such as the x-axis. In our case, this method helps find the volume of the solid created by revolving a region defined by the line and the parabola.

To apply the Disk Method, visualize slicing the solid into thin, circular disks perpendicular to the axis of revolution. Each disk’s radius is determined by the function defining the outer surface, relative to the axis.

• The volume of each disk is calculated using the formula for the area of a circle, \( A = \pi R^2 \), where \(R\) is the radius.
• To find the total volume, integrate these infinitesimal disks along the axis within the defined interval.

This leads to the integral formula for volume: \[ V = \pi \int_a^b [R(x)]^2 \, dx \].

In this exercise, the radius function, determined by the difference between the two curves’ y-values, correctly fits into this formula, ensuring an accurate volume computation.

Intersection Points

Intersection points are vital in problems involving overlapping curves, especially when using calculations for areas and volumes. These points determine where curves meet, effectively marking the boundaries of the region under consideration.

For our problem, these intersection points are obtained by setting the two equations equal and solving simultaneously:

• The line’s equation has been expressed as \(x = 2y\).
• Substituting into the parabola’s equation \(y^2 = 4x\), simplifies to find the common solutions \(y = 0\) and \(y = 8\).

Thus, intersecting points at \((0,0)\) and \((16,8)\) can be calculated after substituting back into the original equations.

These coordinates play a key role in establishing the limits of integration, guiding where the Disk Method will calculate the volume.

Limits of Integration

When dealing with integrations in calculus, establishing the correct limits is crucial. These boundaries control where integration takes place, directing which portions of the function contribute to the overall solution.

In this scenario, the limits are determined by the intersecting points discussed earlier. We integrate from \(x = 0\) to \(x = 16\) because these values contain the region between the line and the parabola.

• Lower Limit: The starting point at \(x=0\) means we begin integrating where the curves first meet.
• Upper Limit: The end point at \(x=16\), another intersection, marks where the evaluation ceases.

Understanding these limits ensures the integration captures all regions necessary to compute the correct volume, without extending unnecessarily.

Integral Calculus

Integral calculus is a branch of mathematics focusing on accumulation, just like what we see when calculating areas, lengths, and volumes. Specifically, in this problem, we're harnessing integral calculus to determine the volume of a solid of revolution.

The Disk Method's reliance on integral calculus comes through needing to evaluate an integral to sum all disk volumes between the defined limits. Using the expanded expression from the Disk Method, \(\left( \sqrt{4x} - \frac{x}{2} \right)^2\), we perform the integration.

• This requires adapting familiar techniques to carefully integrate powers of \(x\).
• Each term in the expansion: \(4x\), \(-2x^{3/2}\), and \(\frac{x^2}{4}\), is integrated separately, producing terms like \(2x^2\), \(-\frac{4}{5}x^{5/2}\), and \(\frac{x^3}{12}\).

After determining these antiderivatives, plug in the limits of integration to find the total volume as a definite integral. This process exemplifies the power of integral calculus in not only theory but in practical applications involving complex shapes and figures.