# Find the area under the standard normal curve between z = 1 and z = 2.

We will use the concept of complex numbers and areas under the curve to find the required area.

## Answer: Area enclosed between the curves = 3 π

Let us see how we use the concept of complex numbers and area under the curve to find the required area.

**Explanation: **

Let us talk about the curve z = 1 first.

The complex number z can be written as x + iy in the the complex form .

Hence , x + iy = 1

If we calculate the magnitude of the complex number z = x + iy then it will be equal to ,

|z| = √(x^{2} + y^{2})

Hence , √(x^{2} + y^{2})^{ }= 1

On squaring both sides we get

x^{2} + y^{2 }= 1

This is a circle with a radius of 1, according to the general circle equation

Similarly lets find out the equation of the curve z = 2 in cartesian form ,

If we calculate the magnitude of the complex number z = x + iy then it will be equal to ,

|z| = √(x^{2} + y^{2})

Hence , √(x^{2} + y^{2})^{ }= 2

On squaring both sides we get

x^{2} + y^{2 }= 4

Similarly, this is a circle with a radius 2.

We have to find the area enclosed between the curve x^{2} + y^{2 }= 4 and x^{2} + y^{2 }= 1,

Area of circle with radius 2 - Area of circle with radius 1

= π(2)^{2} - π(1)^{2}

= 4π - 1π = 3π