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## Area of Parabola

A symmetrical open plane curve formed by the intersection of a cone with a plane parallel to its side is known as Parabola. To calculate the Area of Parabola there is a particular formula that is given in mathematics.

Not only Formulas are given but also many derivations are there to find out the exact formula of area of a Parabola. If you are looking for the Parabola Formula, Calculate Area under Parabola, Derivations of Parabola etc. then you must take a look over here.

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### Area of Parabola

A Parabola is a two-dimensional mirror symmetric curve which is approximately U-SHAPED when oriented as shown in the diagram.

The world ‘parabola’ essentially means ‘applications’ due to its relations to conic sections and its applications.

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Shape Of The Parabola

• If a≠0 , the equation y = ax2 + bx + c is the standard form of a quadratic function and its graph is a parabola.
• If a > 0, the parabola opens upward and the lowest point (called the vertex) yields the minimum value of the function.
• If |a|<1, the parabola opens downward and the highest point (called the vertex) yields the maximum value of the function.
• If  |a|>1 the parabola is narrower than the standard parabola y = xbut if a < 1 then it is wider than the standard parabola.

The x-intercepts and value of y of the parabola are found by solving the equation ax2 + bx + c = 0 .

• The x-value of the vertex may be found using the equation x=−b2a
• The y-value of the vertex may then be found by substituting the x-value into the original equation. The vertical line x=−b2a is called the Axis of symmetry.

The x-intercepts and the vertex can be useful in making a sketch of the parabola.

The vertex of Parabola is the midpoint of the line segment joining the focus and the directrix and is perpendicular to the directrix.

Get Derivation with Examples: Area of Polygon Formula

Parabola Equation

The parabola equation is given by y = 2

Now as we have studied all the term about Parabola, lets take a look over the Area Of Parabola:

AREA OF PARABOLA =

= 2/3 *WIDTH * HEIGHT

= 2/3 * b * a

Area Of Parabola By Integration:

AREA =∫abf(x)dx

=  ∫ab(x2)dx

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Derivation of the Equation of the Parabola

A circle is a locus of points whose distance from a fixed point is a constant. A parabola can also be described as a locus of points whose distance from a fixed point and a fixed line not passing through that point is a constant. An Example of a parabola is shown below.

In the figure below, point F is called the focus of the parabola and line l is called its directrix. The vertex of the parabola is at the origin O and the x-axis the perpendicular bisector of FH. If we take any point P (x,y) on the parabola, draw FP, and draw PQ perpendicular to line  where Q is line l, then the distance between F  and P  and  P  and Q are equal.

Suppose that the coordinates of the focus is (0,p) where P>0 , then the directrix is y=-p(can you see why?).

From here we can see that PQ= ιy+pι and PF=√ x2+ (y-p) 2

Since PQ   is equal to PF

ιy+pι and PF=√ x2+ (y-p)2

Squaring both sides, we have

(y+p)2 = x2+ (y-p)2

y2 + 2py + p2 = x2  + y2 -2py + y2

2py= x2 -2py

4py= x2

Y= (1/4p) x2

This is the equation of the parabola with focus (0,p)  and directrix y= -p

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