Volume by cross-section: ellipse and equilateral triangle cross sections?

In summary, the problem asks for the volume of a solid with a base bounded by the ellipse 4x^2+9y^2=36 and cross sections perpendicular to the x-axis in the shape of equilateral triangles and squares. To solve for the volume, the base of the triangle is calculated as 2(sqrt((-4/9)x^2 + 4)) and the height is found using Pythagoras as sqrt((-4/3)x^2 + 12). The area of the triangle is then expressed as (1/2)(2(sqrt((-4/9)x^2 + 4)))(sqrt((-4/3)x^2 + 12)), which can be simplified further.
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zeion
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Volume by cross-section: ellipse and equilateral triangle cross sections??

Homework Statement



The base of a solid is the region bounded by the ellipse 4x^2+9y^2=36. Find the volume of the solid given that cross sections perpendicular to the x-axis are:
a) equilateral triangles
b) squares


Homework Equations





The Attempt at a Solution



So I'm not really sure how ellipses work.. how can I sketch this ellipse?
Beyond that.. I try to calculate the area of the triangle and then integrate in terms of y so the base is changing according to the ellipse curve.

I write the ellipse as:

y = +/-sqrt((-4/9)x^2 + 4)

So the base of the triangle is 2(sqrt((-4/9)x^2 + 4))
And has that as the length on all side since it is equilateral.
Then I try to find the height using Pythagoras and get

h = +/-sqrt((-4/3)x^2 + 12)

Then now I have the area of the triangle as (1/2)bh, which is =

A = (1/2)(2(sqrt((-4/9)x^2 + 4)))(sqrt((-4/3)x^2 + 12))

Then I can integrate in terms of x.. does that look correct so far?
 
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  • #2
zeion said:
h = +/-sqrt((-4/3)x^2 + 12)
h is only positive.
Other than that, looks good, but you can greatly simplify the last expression.
 
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1. What is "Volume by cross-section"?

"Volume by cross-section" refers to a method of calculating the volume of a three-dimensional object by finding the area of the cross-sections of the object and then multiplying it by the distance between the cross-sections.

2. What is an ellipse cross-section?

An ellipse cross-section is a cross-section of an object that is in the shape of an ellipse, which is a closed, curved shape that resembles a flattened circle.

3. What is an equilateral triangle cross-section?

An equilateral triangle cross-section is a cross-section of an object that is in the shape of an equilateral triangle, which is a three-sided shape with all sides and angles equal in length.

4. How do you calculate the volume using ellipse and equilateral triangle cross-sections?

To calculate the volume using ellipse and equilateral triangle cross-sections, first find the area of the cross-section using the appropriate formula (for example, the area of an ellipse is πab, where a and b are the lengths of the semi-major and semi-minor axes). Then, multiply this area by the distance between the cross-sections to get the volume.

5. What types of objects can be measured using volume by cross-section?

Volume by cross-section can be used to measure the volume of any three-dimensional object, as long as the cross-sections are known and the object has a consistent shape throughout. This method is particularly useful for irregularly shaped objects, as it allows for easier calculation of volume compared to traditional methods.

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