Question

Find the area of the shaded region: r=sqrt(theta) theta between 0 and -3pi/2

Answer 1

`1.519\pi`

Explanation:

For any polar curve area of the region under the curve

`(1)/(2) \int\limits^a_b {r^2} \, dt`

Here r is given as

`r=\theta √(\theta)`

Hence area would be

`(1)/(2) \int\limits^0_(-3\pi)/(2)  {theta}^{(3)/(2) }  \, d\theta \\=(1)/(2) (2)/(5)  {theta}^{(5)/(2) }`

=
`=0.2((3\pi)/(2) )^5\\=1.519\pi`

Answer 2

The equation r = square root of theta can be translated into a semi-circle through the simplification to r^2 = theta. When theta is 0, r is zero. When r is -3pi/2, r2 is equal to 9/4 pi^2. The area of the semi-circle is equal to pi r^2 /2. Hence the area is equal to 34.88 units^2