Math Problem Statement

Find the volume of the solid obtained when the region under the curve y = x√(4 - x²) from x = 0 to x = 2 is rotated about the y-axis.

Solution

The problem provided asks to find the volume of a solid obtained by rotating the region under the curve y=x4x2y = x\sqrt{4 - x^2} from x=0x = 0 to x=2x = 2 about the y-axis. This requires applying the method of cylindrical shells.

Step-by-step solution:

  1. Formula for volume using cylindrical shells:

    When rotating around the y-axis, the volume VV is given by the formula: V=2πabxf(x)dxV = 2\pi \int_{a}^{b} x \cdot f(x) \, dx where f(x)f(x) is the function describing the curve, and aa and bb are the limits of integration (in this case, 00 and 22).

  2. Identify the function and limits:

    • f(x)=x4x2f(x) = x\sqrt{4 - x^2}
    • The limits are from x=0x = 0 to x=2x = 2.
  3. Set up the integral: V=2π02xx4x2dx=2π02x24x2dxV = 2\pi \int_{0}^{2} x \cdot x\sqrt{4 - x^2} \, dx = 2\pi \int_{0}^{2} x^2 \sqrt{4 - x^2} \, dx

  4. Substitute u=4x2u = 4 - x^2:

    • du=2xdxdu = -2x \, dx
    • When x=0x = 0, u=4u = 4.
    • When x=2x = 2, u=0u = 0.

    The integral becomes: V=2π4012u(du)=π04u1/2duV = 2\pi \int_{4}^{0} -\frac{1}{2} \sqrt{u} \, (-du) = \pi \int_{0}^{4} u^{1/2} \, du

  5. Integrate: π04u1/2du=π[23u3/2]04\pi \int_{0}^{4} u^{1/2} \, du = \pi \left[ \frac{2}{3} u^{3/2} \right]_{0}^{4} V=π(23(4)3/223(0)3/2)=π(238)=16π3V = \pi \left( \frac{2}{3} (4)^{3/2} - \frac{2}{3} (0)^{3/2} \right) = \pi \left( \frac{2}{3} \cdot 8 \right) = \frac{16\pi}{3}

So, the volume of the solid is 16π3\frac{16\pi}{3} cubic units.


Would you like further details or have any questions?

Here are some related problems:

  1. What is the volume when rotating the same region around the x-axis?
  2. How would you solve this problem if the curve were rotated around the line x=1x = 1?
  3. Can you calculate the surface area of the solid obtained by this rotation?
  4. How does the volume change if the curve is rotated from x=2x = -2 to x=2x = 2?
  5. What is the interpretation of the integral when rotating a curve about the y-axis?

Tip: When using the cylindrical shells method, always ensure that you set up the integral correctly in terms of the radius and height of the shell.

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Math Problem Analysis

Mathematical Concepts

Calculus
Volume of Solids of Revolution
Cylindrical Shells

Formulas

V = 2π∫(x * f(x)) dx (for cylindrical shells)
u-substitution: u = 4 - x², du = -2x dx

Theorems

Method of Cylindrical Shells
u-Substitution

Suitable Grade Level

College Level (Calculus II)