How do you find the volume of the solid obtained by rotating the region bounded by y=x and $y={x}^{2}$ about the line x=-1?

Makayla Boyd
2022-06-28
Answered

How do you find the volume of the solid obtained by rotating the region bounded by y=x and $y={x}^{2}$ about the line x=-1?

You can still ask an expert for help

Jayce Bates

Answered 2022-06-29
Author has **18** answers

By Shell Method,

$V=2\pi {\int}_{0}^{1}(x+1)(x-{x}^{2})dx=2\pi {\int}_{0}^{1}(x-{x}^{3})dx$

$=2\pi [\frac{{x}^{2}}{2}-\frac{{x}^{4}}{4}{]}_{0}^{1}=2\pi (\frac{1}{2}-\frac{1}{4})=\frac{\pi}{2}$

$V=2\pi {\int}_{0}^{1}(x+1)(x-{x}^{2})dx=2\pi {\int}_{0}^{1}(x-{x}^{3})dx$

$=2\pi [\frac{{x}^{2}}{2}-\frac{{x}^{4}}{4}{]}_{0}^{1}=2\pi (\frac{1}{2}-\frac{1}{4})=\frac{\pi}{2}$

asked 2022-06-29

How do you find the average value of ${x}^{3}$ as x varies between -1 and 2

asked 2022-04-19

What is a general solution to the differential equation y'-3y=5?

asked 2022-06-21

What is the arc length of $f(x)=\mathrm{cos}x-{\mathrm{sin}}^{2}x$ on $x\in [0,\pi ]$?

asked 2021-12-03

What is meant by a specified integral?

asked 2022-05-12

How do you use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by $y=32-{x}^{2}$ and $y={x}^{2}$ revolved about the x=4?

asked 2022-01-02

Evaluate the following integral.

$\int (\frac{5}{{x}^{6}}-4\sqrt{x})dx$

asked 2021-12-26

Evaluate the integral.

$\int \frac{\mathrm{sec}2x}{\mathrm{csc}2x}dx$