(1,1)(2,1)(2,3)
mirror Revolve around y-axis.
solve with Shell and washer
please and thank you!
Made a correction
for washer would
R(x)=2x-1 ?
what about r(x)
To solve this problem using the shell and washer methods, we need to first find the bounds of integration and the formulas for the radius functions. Let's start with the correction for the washer method.
Given the points (1,1), (2,1), and (2,3), we have a region in the xy-plane. To find the bounds of integration, we need to determine the range of x values that form the region.
Looking at the points, we see that the x-coordinate ranges from 1 to 2. Therefore, we will integrate with respect to x over the interval [1,2].
Now, let's determine the radius functions for the shell and washer methods.
For the shell method, the radius function is the distance from the axis of rotation (y-axis) to the curve that forms the cross-section. In this case, the distance is simply the x-coordinate.
So, for the shell method, the radius function is r(x) = x.
For the washer method, we need two radius functions: the outer radius (R(x)) and the inner radius (r(x)). The outer radius is the distance from the axis of rotation to the outer curve, while the inner radius is the distance to the inner curve.
To find the radius functions for the washer method, we can look at the given points. Since the region is formed by a straight vertical line and a horizontal line segment, the outer and inner radii will be constant.
Looking at the points (1,1) and (2,3), we can see that the outer radius extends from the y-axis to the curve formed by the horizontal line segment. Therefore, R(x) = 2.
The inner radius, on the other hand, extends from the y-axis to the vertical line. Since the vertical line is at x = 1, the inner radius will be 1 unit.
So, for the washer method, the outer radius function is R(x) = 2 and the inner radius function is r(x) = 1.
Now that we have the bounds of integration and the radius functions, we can use either the shell or washer method to calculate the volume.
For the shell method, the volume is given by the formula:
V = 2π ∫[a, b] x * f(x) dx
Where a and b are the bounds of integration, x is the radius function (r(x) = x), and f(x) is the height of the region.
For the washer method, the volume is given by the formula:
V = π ∫[a, b] [R(x)^2 - r(x)^2] dx
Where a and b are the bounds of integration, R(x) is the outer radius function (R(x) = 2), and r(x) is the inner radius function (r(x) = 1).
Now, you can plug in the bounds of integration and the respective radius functions to calculate the volume using either the shell or washer method.