What is the Volume of the Solid Using Cylindrical Shells for y=-e^(-x^2)?

Thread starter sheldonrocks97
Start date Jan 29, 2014
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Solid Volume Volume of solid

In summary, the formula for calculating the volume of a solid with the equation y=-e^(-x^2) is ∫0^1(πe^(-2x^2))dx, with integration limits of 0 and 1. The volume can be visualized as a three-dimensional shape with a curved base and a gradually decreasing height, and is directly related to the function e^(-x^2). In real-world applications, the volume of the solid y=-e^(-x^2) has various uses, such as in physics, engineering, and modeling complex systems.

Jan 29, 2014

sheldonrocks97

Gold Member

Homework Statement

Find the volume of the solid using cylindrical shells:

y=e^{-x^2} y=0, x=0, x=1, about y-axis.

Homework Equations

How do I integrate xe^(-x^2)?

The Attempt at a Solution

2∏x∫0 to 1 xe^(-x^2) dx

2∏*-(e^(-1))/2)

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Jan 29, 2014

dirk_mec1

It's actually very easy think of a substitution.

FAQ: What is the Volume of the Solid Using Cylindrical Shells for y=-e^(-x^2)?

What is the formula for calculating the volume of the solid y=-e^(-x^2)?

The formula for calculating the volume of a solid with the equation y=-e^(-x^2) is ∫0^1(πe^(-2x^2))dx. This integral can be solved using various mathematical techniques such as integration by parts or substitution.

What are the limits of integration for finding the volume of the solid y=-e^(-x^2)?

The limits of integration for finding the volume of a solid with the equation y=-e^(-x^2) are 0 and 1. This means that the integral will be evaluated from 0 to 1, with 0 being the lower limit and 1 being the upper limit.

How can the volume of the solid y=-e^(-x^2) be visualized?

The volume of the solid y=-e^(-x^2) can be visualized as a three-dimensional shape with a curved base and a gradually decreasing height. This shape can be difficult to imagine, but it can be visualized by using a graphing calculator or by plotting points on a graph.

What is the relationship between the volume of the solid y=-e^(-x^2) and the function e^(-x^2)?

The volume of the solid y=-e^(-x^2) is directly related to the function e^(-x^2). This means that as the value of e^(-x^2) changes, the volume of the solid will also change. Additionally, the shape of the solid will be determined by the behavior of the function e^(-x^2).

What is the significance of the volume of the solid y=-e^(-x^2) in real-world applications?

The volume of the solid y=-e^(-x^2) has various real-world applications, such as in physics and engineering. For example, it can be used to calculate the volume of a gas in a container with a curved base, or the volume of a curved object in a fluid. It can also be used in modeling the behavior of complex systems in biology and economics.