Math Problem Statement

Q4. Find the volume of the solid bounded by x=7-y^2-z^2 and x=3.

Solution

To find the volume of the solid bounded by the surfaces x=7y2z2x = 7 - y^2 - z^2 and x=3x = 3, we can set up a triple integral.

Step 1: Set Up the Equations

  • The equation x=7y2z2x = 7 - y^2 - z^2 represents a paraboloid opening towards the negative xx-axis.
  • The equation x=3x = 3 represents a plane perpendicular to the xx-axis.

Step 2: Determine the Region of Integration

The region of integration is the solid between these two surfaces.

  1. Intersection of the surfaces: Set 7y2z2=37 - y^2 - z^2 = 3 to find the boundary in the yzyz-plane: 4=y2+z24 = y^2 + z^2 This is a circle in the yzyz-plane with a radius r=2r = 2.

  2. Limits for yy and zz: y2+z24y^2 + z^2 \leq 4 This implies yy and zz are bounded within the circle of radius 2: 2y2,4y2z4y2-2 \leq y \leq 2, \quad -\sqrt{4 - y^2} \leq z \leq \sqrt{4 - y^2}

  3. Limits for xx: For a fixed yy and zz, 3x7y2z23 \leq x \leq 7 - y^2 - z^2

Step 3: Set Up the Integral

The volume VV can be calculated by integrating over the region DD: V=224y24y237y2z2dxdzdyV = \int_{-2}^{2} \int_{-\sqrt{4-y^2}}^{\sqrt{4-y^2}} \int_{3}^{7-y^2-z^2} dx \, dz \, dy

Step 4: Evaluate the Integral

Start by integrating with respect to xx: 37y2z2dx=(7y2z2)3=4y2z2\int_{3}^{7-y^2-z^2} dx = (7 - y^2 - z^2) - 3 = 4 - y^2 - z^2 So the volume integral becomes: V=224y24y2(4y2z2)dzdyV = \int_{-2}^{2} \int_{-\sqrt{4-y^2}}^{\sqrt{4-y^2}} (4 - y^2 - z^2) \, dz \, dy

Step 5: Simplify and Integrate

Convert to polar coordinates:

  • Let y=rcosθy = r \cos\theta and z=rsinθz = r \sin\theta
  • Then, y2+z2=r2y^2 + z^2 = r^2 and the Jacobian of the transformation is rr.

Now, the volume integral is: V=02π02(4r2)rdrdθV = \int_{0}^{2\pi} \int_{0}^{2} (4 - r^2) r \, dr \, d\theta

First, integrate with respect to rr: [ \int_{0}^{2} (4r - r^3) , dr = \left[2r^2 - \frac{r^4}{4}\right]_{0}^{2} = \left[8 - 4\right] = 4 ]

Finally, integrate with respect to θ\theta: V=02π4dθ=4×2π=8πV = \int_{0}^{2\pi} 4 \, d\theta = 4 \times 2\pi = 8\pi

Final Answer:

The volume of the solid is 8π\boxed{8\pi}.


Would you like more details on this process, or do you have any other questions?

Related Questions:

  1. How do you find the surface area of the same solid?
  2. What is the centroid of the given solid?
  3. How would you calculate the volume if the paraboloid had a different equation?
  4. Can the method of cylindrical shells be used to find this volume?
  5. How does changing the plane x=3x = 3 affect the volume?
  6. What happens to the volume if the paraboloid is rotated about the xx-axis?
  7. How does one convert a double integral in Cartesian coordinates to polar coordinates in general?
  8. What is the importance of the Jacobian when converting to polar coordinates?

Tip:

When converting from Cartesian to polar coordinates, remember that the radius rr represents the distance from the origin, and the Jacobian determinant helps account for the area scaling in the new coordinate system.

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Math Problem Analysis

Mathematical Concepts

Triple Integrals
Volume Calculation
Surface Integration

Formulas

Triple integral setup

Theorems

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Suitable Grade Level

Advanced College