let V be volume of the solid obtained by rotating about the y-axis the region bounded by y=sqrtx and y=x^2. find V both by slicing and by cylindrical shells. in both cases, draw a diagram to explain your method.
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To find the volume of the solid obtained by rotating the region bounded by y = √x and y = x^2 about the y-axis, we can use two different methods: slicing and cylindrical shells.
First, let's start with the slicing method.
1. Slicing Method:
To calculate the volume using slicing, we can consider that each infinitesimally thin slice of the solid formed by rotating the region is a disk.
Step 1: Draw a diagram of the region between y = √x and y = x^2 in the xy-plane.
```
^
|
√x |
|
x^2 +---------+
| |
+---------+---->
x-axis
```
Step 2: Determine the limits of integration. In this case, to perform the integration with respect to y, we need to find the y-values at which the two curves intersect. Setting √x = x^2, we have x^4 - x^2 = 0. Solving this quadratic equation, we get x = 0 and x = 1.
Step 3: Express the volume element dV of each slice in terms of y. Since we are rotating about the y-axis, the volume element will be a disk with radius equal to the x-coordinate of the curve y = √x. So, the radius r is given by r = √x.
Step 4: Express the volume element dV in terms of y. Since the volume element of each slice is a disk, its volume is given by dV = πr^2 * dy. Substituting r = √x, we have dV = πx * dy.
Step 5: Integrate to find the total volume. The volume V can be obtained by integrating the volume element over the range of y-values. The integral expression is:
V = ∫[a,b] πx * dy, where a and b are the limits of integration.
Finally, compute the integral to determine the volume V.
2. Cylindrical Shells Method:
To calculate the volume using cylindrical shells, we consider each infinitesimally thin shell as a hollow cylinder.
Step 1: Draw a diagram of the region between y = √x and y = x^2 in the xy-plane.
```
^
|
√x |
|
x^2 +---------+
| |
+---------+---->
x-axis
```
Step 2: Determine the limits of integration. We still need to find the y-values at which the two curves intersect. As mentioned earlier, the curves intersect at y = 0 and y = 1.
Step 3: Express the volume element dV of each shell in terms of y. Since we are rotating about the y-axis, the volume element will be a hollow cylinder. The height of each shell is given by dy, and the radius is the distance between the y-axis and the curve x = √y. Thus, the radius r is given by r = √y.
Step 4: Express the volume element dV in terms of y. The volume of each shell is given by dV = 2πrh * dy, where h represents the thickness of each cylindrical shell.
Step 5: Integrate to find the total volume. The volume V can be obtained by integrating the volume element over the range of y-values. The integral expression is:
V = ∫[a,b] 2πrh * dy, where a and b are the limits of integration.
Evaluate this integral to determine the volume V.
These are the steps to find the volume using both the slicing and cylindrical shells methods. Remember to substitute the appropriate limits of integration and evaluate the integral to obtain the final answers.