If the region enclosed by the straight lines y= x x= 1 and the y= 0 is revolved about the x axis, compute the volume of the solid so generated.

To compute the volume of the solid generated by revolving the region enclosed by the straight lines \( y = x \), \( x = 1 \), and \( y = 0 \) about the \( x \)-axis, we can use the method of disks or washers.

### Step 1: Identify the bounds and the function
The region is bounded by:
- The curve \( y = x \)
- The vertical line \( x = 1 \)
- The horizontal line \( y = 0 \) (which is the \( x \)-axis)

### Step 2: Set up the integral
The volume \( V \) of the solid of revolution about the \( x \)-axis can be found using the formula:
\[
V = \pi \int_{a}^{b} [f(x)]^2 \, dx
\]
where \( f(x) \) is the function being revolved and \( [a, b] \) are the limits of integration.

Here, \( f(x) = x \) and the limits of integration are from \( x = 0 \) to \( x = 1 \).

Thus, the volume is:
\[
V = \pi \int_{0}^{1} (x)^2 \, dx
\]

### Step 3: Compute the integral
First, integrate \( x^2 \):
\[
\int x^2 \, dx = \frac{x^3}{3}
\]

Now, evaluate this integral from 0 to 1:
\[
V = \pi \left[\frac{x^3}{3}\right]_{0}^{1}
\]
\[
V = \pi \left(\frac{1^3}{3} - \frac{0^3}{3}\right)
\]
\[
V = \pi \left(\frac{1}{3} - 0\right)
\]
\[
V = \frac{\pi}{3}
\]

### Final Answer:
The volume of the solid generated by revolving the given region about the \( x \)-axis is \( \frac{\pi}{3} \) cubic units.

Would you like any further details or have any questions? 

Here are 5 related questions you might find interesting:
1. What would the volume be if the region were revolved around the y-axis instead?
2. How would the volume change if the line \( x = 1 \) were replaced with \( x = 2 \)?
3. Can you compute the surface area of the solid generated by the same revolution?
4. What if the curve was \( y = x^2 \) instead of \( y = x \)? How would that affect the volume?
5. How does the method of cylindrical shells compare to the disk method for this problem?

**Tip:** When setting up integrals for volumes of revolution, always carefully sketch the region and identify the axis of rotation to ensure the correct bounds and functions are used.

Learn how to compute the volume of a solid generated by revolving the region enclosed by the lines y = x, x = 1, and y = 0 about the x-axis. Detailed step-by-step solution using the method of disks or washers. Suitable for advanced high school students studying calculus.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation