4) when the area bounded by y = 0, y = e^x - 1 and x = 3 is rotated around the y - axis the resulting cross…

Explanation:

Step1: Analyze the region and rotation

We have the curves $y = 0$, $y=e^{x}-1$ and $x = 3$. When rotating about the $y -$axis, we use the method of cylindrical - shells or the washer method. First, we need to express $x$ in terms of $y$ for $y=e^{x}-1$, so $x=\ln(y + 1)$. The right - most $x$ value in the region is $x = 3$ and the left - most $x$ value (in terms of $y$) is $x=\ln(y + 1)$ for $y\geq0$.

Step2: Determine the cross - section type

When rotating a region about the $y$-axis, for a given $y$ value, the outer radius $R$ of the cross - section is the distance from the $y$-axis to the right - most boundary of the region and the inner radius $r$ is the distance from the $y$-axis to the left - most boundary of the region. The outer radius $R = 3$ and the inner radius $r=\ln(y + 1)$. The cross - sections are washers.

Answer:

b) Washers with a big radius of 3 and a little radius of $\ln(y + 1)$