Table of Contents
- 1 How do you find the area under a curve using integration?
- 2 How did Archimedes determine the area of a circle?
- 3 How do you explain the area of a circle?
- 4 How did Archimedes calculate pi?
- 5 What does the area under the curve represent statistics?
- 6 How did Archimedes calculate the area of a sphere?
- 7 How do you find the area under a curve in math?
- 8 What is Archimedes’ theorem?
How do you find the area under a curve using integration?
The area under a curve between two points can be found by doing a definite integral between the two points. To find the area under the curve y = f(x) between x = a and x = b, integrate y = f(x) between the limits of a and b. Areas under the x-axis will come out negative and areas above the x-axis will be positive.
How did Archimedes determine the area of a circle?
The way Archimedes formulated his Proposition about the area of a circle is that it is equal to the area of a triangle whose height is equal to it radius and whose base is equal to its circumference: (1/2)(r · 2πr) = πr2.
What does the area under the curve represent physics?
The area under the curve is the magnitude of the displacement, which is equal to the distance traveled (only for constant acceleration).
How do you explain the area of a circle?
The area of a circle is pi times the radius squared (A = π r²).
How did Archimedes calculate pi?
Archimedes’ method finds an approximation of pi by determining the length of the perimeter of a polygon inscribed within a circle (which is less than the circumference of the circle) and the perimeter of a polygon circumscribed outside a circle (which is greater than the circumference).
What does area under a curve represent in calculus?
The area under the curve is defined as the region bounded by the function we’re working with, vertical lines representing the function’s bounds, and the -axis. The graph above shows the area under the curve of the continuous function, . The interval, , represents the vertical bounds of the function.
What does the area under the curve represent statistics?
The area under the normal distribution curve represents probability and the total area under the curve sums to one. Most of the continuous data values in a normal distribution tend to cluster around the mean, and the further a value is from the mean, the less likely it is to occur.
How did Archimedes calculate the area of a sphere?
The problem arose first in what is perhaps the most famous work of Archimedes, On the sphere and cylinder I, in which he calculates the area of a sphere by comparing it to an enclosing cylinder. This is a more complicated notion that that of the length of the circumference of a circle, but it is similar.
How can the Archimedes principle be used to determine the density?
How can the Archimedes Principle be used to determine the density? The weight of the fluid displaced is equal to the buoyant force on a submerged object. The mass divided by the volume thus determined gives a measure of the average density of the object. Stay tuned with BYJU’S to learn more interesting topics with engaging videos!
How do you find the area under a curve in math?
Sometimes the only possible way is to sum vertically. x y d c x. dy. The best way to find the area under this curve is by summing vertically. In this case, we find the area is the sum of the rectangles, heights. x = f ( y) \\displaystyle {x}= f { {\\left ( {y}\\right)}} x= f (y) and width.
What is Archimedes’ theorem?
The statement of Archimedes’ Theorem. In an algebraic formulation, we say that the area of a circle is πr2 and its circumference is 2πr. These are consistent with Archimedes’ claim: πr2 = (1/2)⋅r⋅ (2πr). But the ancient Greeks did not have algebra, and they did not have the notion of a real number that we do.