# Area and volume calculation (no integration))

• Jhenrique
In summary, the speaker discusses how to compute the area of a rectangle formed by Δx and Δy, as well as the volume of a parallelepiped formed by Δx, Δy, and Δz. They also mention the challenge of computing the volume in spherical coordinates and suggest using a formula derived from calculus or the ancient Greek method of exhaustion.
Jhenrique
I can compute the area of the rectangle formed by Δx and Δy simply by product ΔxΔy.

Now, how can I to compute the area in gray given Δr and Δθ?

Also, I can to compute the volume of a parallelepiped formed by Δx, Δy and Δz, simply multiplicand ΔxΔyΔz. But, how can I compute the volume in shpherical coordinates formed by Δρ, Δφ and Δθ?

Convention:

If you know the radius then you can compute the wedge by computing the area of circles r and r+dr and an angle ratio dtheta/(2*pi) for dtheta measured in radians:

area(wedge) = area(circle(r+dr) - area(circle(r)) * (dtheta / 2*pi)

In spherical coordinates you have a more complicated situation where you could derive a formula for the 3D wedge using calculus.

If you're thinking the ancient greeks didn't know calculus so how did they algebraically arrive at the answer then
you need to read about their calculus precursor the method of exhaustion:

http://en.wikipedia.org/wiki/Method_of_exhaustion

http://www.math.ubc.ca/~cass/courses/m446-03/exhaustion.pdf

http://www-groups.dcs.st-and.ac.uk/history/HistTopics/The_rise_of_calculus.html

## 1. What is the difference between area and volume?

Area is the measure of the surface of a two-dimensional shape, while volume is the measure of the space occupied by a three-dimensional object.

## 2. How do you calculate the area of a shape?

The formula for calculating the area of a shape depends on its shape. For example, the area of a square can be calculated by multiplying the length of one side by itself, while the area of a circle can be calculated by multiplying pi (3.14) by the radius squared.

## 3. What is the formula for finding the volume of a solid object?

The formula for calculating the volume of a solid object also depends on its shape. For example, the volume of a cube can be calculated by multiplying the length, width, and height, while the volume of a cylinder can be calculated by multiplying pi (3.14) by the radius squared and the height.

## 4. Can you estimate the area or volume of an irregular shape?

Yes, the area and volume of an irregular shape can be estimated by breaking it down into smaller shapes that can be easily calculated, such as triangles or rectangles, and then adding their areas or volumes together.

## 5. Why is it important to know how to calculate area and volume?

Knowing how to calculate area and volume is essential in fields such as engineering, architecture, and construction. It allows us to accurately measure and estimate the amount of materials needed for a project, as well as determine the capacity and strength of structures.

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