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 and , we can set up a triple integral.
Step 1: Set Up the Equations
- The equation represents a paraboloid opening towards the negative -axis.
- The equation represents a plane perpendicular to the -axis.
Step 2: Determine the Region of Integration
The region of integration is the solid between these two surfaces.
-
Intersection of the surfaces: Set to find the boundary in the -plane: This is a circle in the -plane with a radius .
-
Limits for and : This implies and are bounded within the circle of radius 2:
-
Limits for : For a fixed and ,
Step 3: Set Up the Integral
The volume can be calculated by integrating over the region :
Step 4: Evaluate the Integral
Start by integrating with respect to : So the volume integral becomes:
Step 5: Simplify and Integrate
Convert to polar coordinates:
- Let and
- Then, and the Jacobian of the transformation is .
Now, the volume integral is:
First, integrate with respect to : [ \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 :
Final Answer:
The volume of the solid is .
Would you like more details on this process, or do you have any other questions?
Related Questions:
- How do you find the surface area of the same solid?
- What is the centroid of the given solid?
- How would you calculate the volume if the paraboloid had a different equation?
- Can the method of cylindrical shells be used to find this volume?
- How does changing the plane affect the volume?
- What happens to the volume if the paraboloid is rotated about the -axis?
- How does one convert a double integral in Cartesian coordinates to polar coordinates in general?
- What is the importance of the Jacobian when converting to polar coordinates?
Tip:
When converting from Cartesian to polar coordinates, remember that the radius represents the distance from the origin, and the Jacobian determinant helps account for the area scaling in the new coordinate system.
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Triple Integrals
Volume Calculation
Surface Integration
Formulas
Triple integral setup
Theorems
-
Suitable Grade Level
Advanced College
Related Recommendation
Calculate Volume of Solid Bounded by x=8-y^2-z^2 and x=y^2+z^2
Find the Volume of a Solid Bounded by Planes in the First Octant
Volume of a Solid in the First Octant Bounded by y+z=3, y=x^2
Iterated Integral Setup for Volume of Solid Bounded by z = 3x and z = x^2
Find the Volume of a Solid in the First Octant Bounded by Two Planes