How do you find the volume of the solid formed by rotating the region enclosed by ?

about the x-axis.

Question

about the x-axis.

Isabella Ackerman · Accepted Answer

Awesome Ken...

Here is a Maple rendering of the rotation!

By the way, you have the coolest name in the world.

Scarlett Allison · Answer

0.9117

#f(r)=e^x# #V(R)=2 pi int_0^R f(r)rdr# #C(h,r)=pi h r^2# now with #h = 2# and #R = 0.3# #V_t =V(R) + C(2,R) = 0.3462+0.5655 = 0.9117#

Christopher Agresta · Answer

#V=pi(e^(0.6)/2+4e^0.3-3.3)~~9.4577...#

We'll use a method call the disk method to do this.

Luke Alvarez · Answer

undefined To find the volume of the solid formed by rotating a region enclosed by a curve about an axis, you can use the method of cylindrical shells or disk/washer method depending on the shape of the region and the axis of rotation.

1. **Cylindrical Shells Method**:
   - For regions enclosed by curves that are parallel to the axis of rotation, you can use the cylindrical shells method.
   - The formula for the volume of the solid generated by cylindrical shells is given by:
     $$ V = \int_{a}^{b} 2\pi x \cdot f(x) \, dx $$
   - Where $ f(x) $ represents the height of the shell at position $ x $, and $ a $ and $ b $ are the limits of integration.

2. **Disk/Washer Method**:
   - For regions enclosed by curves that are perpendicular to the axis of rotation, you can use the disk or washer method.
   - The formula for the volume of the solid generated by disks or washers is given by:
     $$ V = \pi \int_{a}^{b} [f(x)]^2 \, dx $$
   - Where $ f(x) $ represents the distance from the curve to the axis of rotation at position $ x $, and $ a $ and $ b $ are the limits of integration.

To use these methods, you need to:
- Identify the axis of rotation.
- Determine whether the curves enclosing the region are parallel or perpendicular to the axis of rotation.
- Choose the appropriate method accordingly.
- Set up the integral based on the method chosen.
- Evaluate the integral to find the volume.