AREA BOUNDED BY A CURVE AND THE Y AXIS

Let x = f(y) be a continuous function of y on [c, d] to the right of y-axis as shown below.

Then, the area bounded by the curve x = f(y) and the y-values y = c and y = d to the right of y-axis is given by

What if the curve lies to the left of y-axis between the lines y = c and y = d?

Then, the area bounded by the curve x = f(y) and the y-values y = c and y = d to the left of y-axis is given by

Example 1 :

Find the area of the region bounded by y = 2x + 1, y = 3, y = 5 and y–axis.

Solution :

The line y = 2x + 1 lies to the right of y-axis between the lines y = 3 and y = 5 as shown below.

Solve the equation y = 2x + 1 for x in terms of y.

y = 2x + 1

y - 1 = 2x

⁽ʸ ⁻ ¹⁾⁄₂ = x

The required area :

Example 2 :

Find the area of the region bounded by y = 2x + 4, y = 1, y = 3 and y–axis.

Solution :

The line y = 2x + 4 lies to the left of y-axis between the lines y = 1 and y = 3 as shown below.

Solve the equation y = 2x + 4 for x in terms of y.

y = 2x + 4

y - 4 = 2x

⁽ʸ ⁻ ⁴⁾⁄₂ = x

The required area :

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