The Washer Method

When calculating the area under the curve, it may be easier to visualize splitting the curve into multiple rectangles. Then finding the area of each of those rectangles to sum them together for the total area. However, once these curves rotate around an axis, they become a 3D shape and the same struggle to find the volume occurs. Similarly, to calculate the volume of this type of solid, it may be easier to break it down into multiple flat or standardized disks (or if there is a hole they are known as washers). Then finding the volume of each disk (or washer) and sum together for the total volume. This would be known as the washer method.

Choosing a Basic Function

To best understand this concept, I chose a basic function $f(x)=x^{2}$ I chose this function because it is important to completely understand how a method works on the most standard equation before moving onto more complicated functions. Even though this is not the best example because it only shows the change in cylinder size inside the object, it still shows a good representation of how volume can be calculated while focusing only on one part.

Calculating Volume

Approximate Volume

Like we learned earlier, to find the approximate volume we will use the washer method. Therefore, we need to calculate the volume of each disk/washer in the object.

First, we must calculate the height of each washer. The interval is 1.5 inches long with 10 washers located inside that interval.

H = 1.5 / 10 = 0.15 inches

Heights are constant for each washer, so each washer will have a height of 0.15 inches.

Next, we will need to determine the radius for each washer. To do this we will take the outer radius and subtract the inner radius. The outer radius will always be 1.5 inches because there is a smooth curve on the outside of the object. For the inner radius we will use the function $x^{2}$ in terms of y (because the object is rotated about the y-axis), so $^{\sqrt{y}}$ . This leaves us with the following equation for each washer

$V = \prod (outer radius^{2}-inner radius^{2})(height)$

The following table calculates the volumes of every disk and washer.

Therefore, the total estimation of volume for this object is $7.422 in^{3}$ .

However, this is just an estimation because it is not a smooth curve.

Volume

To find a more accurate volume, we can take the integral of the curve. This would account for the smooth edges of the object compared to the sharp change in edges of the washers. The formula is the following,

$V =\prod \int_{0}^{2.25}{R(y)^{2}-{r(y)^{^{2}}}}dy$

Continuing this equation with the dimensions of our object

$V =\prod \int_{0}^{2.25}{1.5^{2}-{\sqrt{y}^{^{2}}}}dy$

$8.8357 in^{3}$

These volumes are close together proving that the first was just an estimation.

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