Find the volume of the solid generated by revolving the region bounded by $y = x³, y = 0$ and $x = 2$ about the y-axis using the washer method.

Step 1: Express x in terms of y

To use the washer method when revolving around the $y$ -axis, we need to express the functions in terms of $y$. The given curve is $y = x^3$. Taking the cube root of both sides, we get $x = y^{1/3}$. The other boundary is $x=2$, which remains $x=2$.

Step 2: Determine the limits of integration

The region is bounded by $y=0$ (the $x$ -axis) and $x=2$. When $x=2$, $y = 2^3 = 8$. So, the limits of integration for $y$ are from $0$ to $8$.

Step 3: Identify the outer and inner radii

When revolving about the $y$ -axis, the outer radius $R(y)$ is the distance from the $y$ -axis to the outer boundary, which is $x=2$. The inner radius $r(y)$ is the distance from the $y$ -axis to the inner boundary, which is $x = y^{1/3}$.

Step 4: Set up the integral for the volume

The formula for the volume using the washer method about the $y$ -axis is $V = \pi \int_{a}^{b} ([R(y)]^2 - [r(y)]^2) dy$. Substituting the radii and limits, we get the integral.

Step 5: Evaluate the integral

Now, we evaluate the definite integral. We find the antiderivative of $4 - y^{2/3}$, which is $4y - \frac{y^{5/3}}{5/3}$. Then, we evaluate this expression at the upper and lower limits and subtract.

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