Finding volume using integration

Thread starter ver_mathstats
Start date Nov 23, 2019
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Integration Volume

In summary, integration is a mathematical process used to find the area or volume of a shape or object by breaking it down into smaller pieces and summing their individual areas. It is used to find volume by breaking down a three-dimensional object into thin slices and finding their areas using integration. The formula for finding volume using integration is ∫a^b A(x) dx, and the two main types of integration used for volume calculation are single and double integration. Some real-life applications of finding volume using integration include engineering, architecture, and physics, where it is used to calculate liquid volumes, material quantities, and object properties.

Nov 23, 2019

ver_mathstats

Homework Statement: The base of the solid is a square, one of whose sides is the interval [0,5] along the the x-axis.
The cross sections perpendicular to the x-axis are rectangles of height f(x)=4x^2. Compute the volume of the solid.

Relevant Equations: f(x)=4x^2

I know that the formula for volume is equal to the definite integral ∫A(x)dx, where A(x) is the cross sectional. I found the definite integral where b=5 and a=0, for ∫4x²dx. I obtained the answer 500/3, however this was incorrect, and I'm unsure of where I went wrong?

Thank you.

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Nov 23, 2019

Orodruin

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You have not integrated the area, you have integrated the height.

1. What is the concept of finding volume using integration?

The concept of finding volume using integration is based on the mathematical principle that the volume of a 3-dimensional object can be calculated by integrating the cross-sectional area of the object over a given interval. This method is commonly used in calculus to find the volume of irregularly shaped objects or objects with varying cross-sectional areas.

2. How is integration used to find volume?

To find volume using integration, the object is divided into infinitesimally thin slices, and the volume of each slice is calculated using integration. The volume of the entire object is then obtained by adding up the volumes of all the slices. This process is also known as the method of disks or washers.

3. What are the steps involved in finding volume using integration?

The steps involved in finding volume using integration are:

Identify the shape of the object and the limits of integration.
Choose a cross-sectional area formula that represents the shape of the object.
Express the cross-sectional area in terms of the variable of integration.
Integrate the cross-sectional area over the given interval.
Evaluate the definite integral to obtain the volume of the object.

4. What are some real-life applications of finding volume using integration?

Finding volume using integration has various real-life applications, such as calculating the volume of irregularly shaped objects in engineering and architecture, determining the volume of fluids in containers or pipes, and estimating the amount of material needed for construction or manufacturing.

5. Are there any limitations to using integration for finding volume?

While integration is a powerful tool for finding volume, it does have some limitations. It is not suitable for finding the volume of objects with changing cross-sectional areas along the same axis, such as a cone or pyramid. Integration also assumes that the object is continuous and has a smooth surface, which may not always be the case in real-life scenarios.