Question

Using integration, find the area bounded by the tangent to the curve 4y = x² at the point (2, 1) and the lines whose equations are x = 2y and x = 3y – 3.

Answer 1

Obviously 4y = x² is upward parabola having vertex at origin.

Now 4y = x²

⇒ Slope of tangent at (2, 1) to given curve 4y = x² is 1.

Equation of tangent \(=\frac{y-1}{x-2}=1\)

⇒ y - 1 = x - 2 ⇒ y = x -1

Now, for graph of x = 2y

x	0	2
y	0	1

Also for graph of x = 3y -3

x	0	3
y	1	2

After plotting the graph, we get shaded region ABC as required region, area of which is to be calculated.

After solving the respective equation, we get

Coordinate of A ≡ (2, 1); B ≡ (6, 3); C ≡ (3, 2)

Now, the required area = area of shaded region ABC

= ar(region ALMC) + (region CMNB) - ar(region ALNB)