Determining the Area Between Parabolas y24x and x24y
Understanding the area between the curves of the parabolas y24x and x24y involves the application of integral calculus. In this article, we will explore the steps to calculate this area and discuss the geometric properties and symmetries of these parabolas.
Introduction to Parabolas and Intersection Points
Parabolas are curves where any point is at an equal distance from a fixed point (the focus) and a fixed straight line (the directrix). The equations given above,y24x and x24y, define two distinct parabolas.
Calculating the Intersection Points
To find the intersection points of the two parabolas, we equate the expressions:
y24x x24y
Using the substitution yx2/4 in the first equation, we get:
(x2/4)2 4x
x?/16 4x
x? - 64x 0
x(x3 - 64) 0
This simplifies to:
x(x - 4)(x2 4x 16) 0
The roots are:
x 0, 4, and two complex roots (irrelevant for this problem).
For x 0, y 0.
For x 4, y 4.
Hence, the two parabolas intersect at (0,0) and (4,4).
Calculating the Area Under the Parabolas
The area under the curve x24y from 0 to 4 is:
A∫04(x2/4)dx
(x3/12)04
43/12 - 0
16/3 square units.
Calculating the Required Area
The area between the two parabolas can be found by subtracting the area under each parabola from the area of the square with side length 4.
Area of square 42 16 square units.
Required area 16 - 2 × (16/3)
16 - 32/3
(48 - 32)/3
16/3 square units.
It is noteworthy that the two curves are symmetric about yx. This symmetry helps in understanding the distribution of the area.
Graphical Representation
The attached graph for reference is courtesy of Desmos. The graph visually represents the intersection points and the intersection area.
For a detailed understanding and visual representation, please refer to the provided graph.
Understanding the area between the curves of these parabolas not only enhances geometrical insight but also showcases the power of integration in solving complex problems. Whether you are a student learning calculus or a professional using calculus in practical applications, this understanding is valuable for problem-solving and research.