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Prove that the curves y^2 = 4x and x^2 = 4y divide the area of square bounded by x = 0, x = 4, y = 4 and y = 0 into three equal parts. - CBSE (Science) Class 12 - Mathematics

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Question

Prove that the curves y2 = 4x and x2 = 4y divide the area of square bounded by x = 0, x = 4, y = 4 and y = 0 into three equal parts.

Solution

A1, A2, Aare the areas denoted in the figure. We need to prove A1 = A= A3.

`A_1 = ∫_0^4y1dx`

`= ∫_0^4x^2/4dx        `

`=1/4[x^3/3]_0^4`

`=16/3 sq. units`

 

`A_2=∫_0^4(y_2−y_1)dx`

`=∫_0^4(sqrt(4x)−x^2/4)dx`

`= [4/3x^3/2−x^3/12]_0^4`

`=16/3sq. units`

`A_3=area bounded by y^2=4x, y=0 and y=4`

`=∫_0^4x_1dy=∫_0^4y^2/4dy=1/4[y^3/3]_0^4=16/3`

`therefore A_1=A_2 =A_3 =16/3 sq. units`

Thus, y2 = 4x and x2 = 4y divide the area of square bounded by x = 0, x = 4, y = 4 and y = 0 into three equal parts.

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Solution Prove that the curves y^2 = 4x and x^2 = 4y divide the area of square bounded by x = 0, x = 4, y = 4 and y = 0 into three equal parts. Concept: Area of the Region Bounded by a Curve and a Line.
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