Math Problem Statement
The region bounded by the given curves is rotated about the specified axis. Find the volume V of the resulting solid by using shell method y = −x2 + 17x − 72, y = 0; about the y−axis
Solution
To find the volume of the solid formed by rotating the region bounded by and around the y-axis using the shell method, we follow these steps:
1. Identify the curves and bounds
We are given the curve:
and the horizontal axis as the bounds of the region.
We need to find where the curve intersects the x-axis, which will give us the limits for . This happens when .
Solve the quadratic equation for :
Using the quadratic formula:
Thus, the solutions are:
So, the region is bounded between and .
2. Set up the volume integral using the shell method
The formula for the volume of a solid of revolution using the shell method is:
Here:
- The radius is the distance from the axis of rotation (the y-axis), which is simply .
- The height is the value of the function, which is .
So, the volume integral becomes:
3. Evaluate the integral
First, expand the integrand:
Now, integrate term by term:
4. Compute the definite integral
Now, let's substitute the limits of integration:
- When :
- When :
Now subtract the values:
Thus, the volume is approximately:
Conclusion:
The volume of the solid formed by rotating the region around the y-axis is approximately cubic units.
Would you like to go over any part in more detail, or need help with a different concept?
Here are 5 related questions to expand your understanding:
- How does the shell method differ from the disk/washer method in finding volumes?
- How would the problem change if the region was rotated about a different axis, like the x-axis?
- What is the significance of the limits of integration in determining the volume?
- Can the volume of the same region be computed using the disk method? How?
- How does the geometry of the region affect the complexity of the volume calculation?
Tip: When solving integrals in volume problems, break down the integrand carefully before attempting to integrate, to avoid mistakes!
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Math Problem Analysis
Mathematical Concepts
Volume of Solids of Revolution
Shell Method
Quadratic Equations
Definite Integrals
Formulas
Volume using Shell Method: V = 2π∫[a,b](radius)(height)dx
Quadratic Formula: x = [-b ± √(b² - 4ac)] / 2a
Theorems
Fundamental Theorem of Calculus
Quadratic Formula
Suitable Grade Level
Undergraduate Calculus
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