# Riemanns Sum Problem: Find the exact volume

• MHB
• MathsKid007
In summary, the exact volume of any shape with hexagonal horizontal cross-sections, can be found using the formula $V = \frac{\sqrt3}2 \int_0^h (r^2 - x^2) \, dx = \frac{\sqrt3}2(h^3 - \frac{r^3}3)$.
MathsKid007
View attachment 8261

This is what i have so far
We can find the exact volume of any shape using:
V= $$\displaystyle int[a,b] A(x) dx$$
Where,A(x)is the cross-sectional area at height x
and [a,b] is the height interval
We know that the horizontal cross-sections are hexagonal
$$\displaystyle ∴A=(3√3)/2 a^2$$
Where a,is the length of a side
Write the side length a,at height x
a= s
$$\displaystyle ∴A=(3√3)/2 s^2$$
$$\displaystyle V= int[0,h](3√3)/2 x^2 dx$$
$$\displaystyle V= (3√3)/2 int[0,h]x^2 dx$$
$$\displaystyle = (3√3)/2*x^3/3$$
$$\displaystyle =[(√3 x^3)/2] [0,h]$$
$$\displaystyle V=(√3 h^3)/2$$
Is this correct?

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MathsKid007 said:
This is what i have so far
We can find the exact volume of any shape using:
V= $$\displaystyle int[a,b] A(x) dx$$
Where,A(x)is the cross-sectional area at height x
and [a,b] is the height interval
We know that the horizontal cross-sections are hexagonal
$$\displaystyle ∴A=(3√3)/2 a^2$$
Where a,is the length of a side
Write the side length a,at height x
a= s
$$\displaystyle ∴A=(3√3)/2 s^2$$
$$\displaystyle V= int[0,h](3√3)/2 x^2 dx$$
$$\displaystyle V= (3√3)/2 int[0,h]x^2 dx$$
$$\displaystyle = (3√3)/2*x^3/3$$
$$\displaystyle =[(√3 x^3)/2] [0,h]$$
$$\displaystyle V=(√3 h^3)/2$$
Is this correct?
Hi MathsKid, and welcome to MHB!

I agree with your solution up to the point $A = \frac{3\sqrt3}2s^2$. But you have gone wrong in the next line, when you form the integral. You need to apply Pythagoras to write $s^2 = r^2 - x^2$, so that $A = \frac{3\sqrt3}2(r^2 - x^2)$. Then you can integrate that from $x=0$ to $x=h$.

## 1. What is Riemann's Sum Problem?

Riemann's Sum Problem is a mathematical problem that involves finding the exact volume of a three-dimensional shape by dividing it into smaller, simpler shapes and then adding up the volumes of those shapes.

## 2. How do you solve Riemann's Sum Problem?

To solve Riemann's Sum Problem, you first need to divide the shape into smaller sections, such as rectangles or cubes. Then, you calculate the volume of each section using the appropriate formula, and finally, you add up all the volumes to find the exact volume of the shape.

## 3. What is the importance of Riemann's Sum Problem?

Riemann's Sum Problem is important in mathematics because it allows us to find the exact volume of complex shapes that cannot be easily calculated using traditional methods. It is also a fundamental concept in calculus and is used to solve various real-world problems involving volume.

## 4. What is the difference between a left and right Riemann sum?

The difference between left and right Riemann sums lies in the way the shape is divided into smaller sections. In a left Riemann sum, the left endpoint of each section is used to calculate the volume, while in a right Riemann sum, the right endpoint is used. This can result in slightly different values for the exact volume, but as the number of sections increases, the difference becomes negligible.

## 5. Are there any limitations to using Riemann's Sum to find the exact volume?

While Riemann's Sum is a powerful tool for finding the exact volume of complex shapes, it does have some limitations. It can only be used for shapes that can be divided into smaller, simpler sections, and the accuracy of the result depends on the number of sections used. Additionally, it may not be the most efficient method for finding the volume in certain situations.

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