Question

region R is revolved about the​ y-axis. The volume of the resulting solid could​ (in principle) be found by using the​ disk/washer method and integrating with respect to

Answer 1

Answer:

A region R is revolved about the y-axis. The volume of the resulting solid could (in principle) be found using the disk/washer method and integrating with respect to y or using the shell method and integrating with respect to x.

Step-by-step explanation:

We assume this question :

Fill in the blanks: A region R is revolved about the y-axis. The volume of the resulting solid could (in principle) be found using the disk/washer method and integrating with respect to _ or using the shell method and integrating with respect to _____.

We can calculate the volume of a region when revolved about y-axis with two common methods: washer method and shell method. We need to take in count, that the general formula for these methods are different respect to the variable used to calculate the volume.

Since the region R is revolved about y-axis, disk/washer method needs to calculate the integral respect to y, and by the other hand the shell method will calculate the integral respect to x.

Washer method

$V\approx \sum_{k=1}^n \pi r^2 h = \sum_{k=1}^n \pi f(y_k)^2 dy$

$V= \pi \int_{c}^d f(y)^2 dy$

Shell method

$V = \lim_{n\to\infty} 2\pi x_i f(x_i)\Delta x =\int_{a}^b 2\pi x f(x) dx$

So then the correct answer would be:

A region R is revolved about the y-axis. The volume of the resulting solid could (in principle) be found using the disk/washer method and integrating with respect to y or using the shell method and integrating with respect to x.