The video explains the application of the washer method in calculus to find the volume of a solid of revolution. It uses the example of revolving the area between the functions $y=x$ and $y=\sqrt{x}$ around the x-axis. 

At any given point $x$, a slice perpendicular to the x-axis would form a washer. The outer radius of this washer is determined by the function further from the axis of revolution, which is $y=\sqrt{x}$, and the inner radius is determined by the function closer to the axis, which is $y=x$. 

The area of a single washer is calculated by subtracting the area of the inner circle from the area of the outer circle: $A = \pi (R_{outer}^2 - R_{inner}^2)$. In this specific example, $R_{outer} = \sqrt{x}$ and $R_{inner} = x$, so the area of one washer is $A = \pi ((\sqrt{x})^2 - x^2) = \pi(x - x^2)$.

To find the total volume of the solid, we need to sum the areas of all these infinitesimally thin washers from the starting point to the ending point of the region. This summation is achieved through integration. For the given region between $x=0$ and $x=1$, the volume is found by integrating the area function: $V = \int_{0}^{1} \pi(x - x^2) dx$.

The video concludes by stating that the result of this integration is a volume of $\pi/6$.

Math With Allison

Why do we use the washer method? Let's take the area between x and the square root of x and let's revolve it around the x-axis.

So here, if we took a slice, we would have a washer. We'd have a smaller circle with a smaller radius, and this would extend out to the function x. And then we'd have a larger radius, and this would extend all the way out to the function the square root of x.

So how do we find the area of this? Well, we take the area of the larger circle and we subtract the area of the smaller circle. So here we get pi * x - x^2.

Now, this is for only one of those slices, and we want to take all the slices between 0 and 1. So this is where the integral comes in. We're going to take all the washers between 0 and 1 and add up their areas together.

So here, when we integrate, we end up with the volume of pi over six.

Washer Method

why do we use the washer method let's
take the area between X and square root
of x and let's revolve it around the
xaxis so here if we took a slice we
would have a washer we'd have a smaller
circle with a smaller radius and this
would extend out to the function X and
then we'd have a larger radius and this
would extend all the way out to the
function the square root of x so how do
we find the area of this well we take
the area of the larger Circle and we
subtract the area of the smaller Circle
so here we get Pi * x - x^2 now this is
for only one of those slices and we want
to take all the slices between 0 one so
this is where the integral comes in
we're going to take all the washers
between 0 and one and add up their areas
together so here when we integrate we
end up with the volume of pi over six

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Washer Method

Transcription