Math Problem Statement
find the volume of the solid obtained when the region under the curve y=xsqrt(4-x^2) from x=0 to x=2 is rotated about the y-axis.
Solution
To find the volume of the solid obtained by rotating the region under the curve from to around the y-axis, we use the method of cylindrical shells.
The formula for the volume of a solid generated by revolving a region around the y-axis is:
Here, , and the limits of integration are to .
Thus, the volume is given by:
Simplify the integrand:
Step 1: Substitution
We perform the substitution:
When , , and when , .
Rewriting the integral in terms of :
Flipping the limits and simplifying:
Step 2: Distribute and Integrate
Now, distribute to each term:
This becomes:
Now, integrate each term:
Step 3: Evaluate the Integrals
Now substitute back the limits of integration:
Step 4: Combine the Results
Now subtract the integrals:
To combine the terms, get a common denominator:
Finally:
Thus, the volume of the solid is .
Do you want further details on the steps? Here are 5 related questions:
- How would you calculate the volume if the region was rotated around the x-axis?
- Can this method be used for curves that intersect the axis of rotation?
- How does the cylindrical shell method compare to the disk method?
- What is the geometric significance of the substitution used in this problem?
- Can the integral limits change if the curve is shifted vertically?
Tip: When rotating regions, always visualize the axis of rotation to choose the right method: shells (for rotation around a vertical axis) or disks (for rotation around a horizontal axis).
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Math Problem Analysis
Mathematical Concepts
Cylindrical Shell Method
Definite Integration
Substitution Method
Solid of Revolution
Formulas
Volume of a solid using cylindrical shells: V = 2π ∫[a,b] x * f(x) dx
Substitution: u = 4 - x^2, du = -2x dx
Integrals of powers of u: ∫u^(1/2) du = (2/3) u^(3/2), ∫u^(3/2) du = (2/5) u^(5/2)
Theorems
Fundamental Theorem of Calculus
Substitution in Definite Integrals
Suitable Grade Level
College Level - Calculus II
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