Question

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

Answer 1

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

Answer 2

Answer:

$1.519\pi$

Step-by-step explanation:

For any polar curve area of the region under the curve

$\frac{1}{2} \int\limits^a_b {r^2} \, dt$

Here r is given as

$r=\theta \sqrt{\theta}$

Hence area would be

$\frac{1}{2} \int\limits^0_\frac{-3\pi}{2} {theta}^{\frac{3}{2} } \, d\theta \\=\frac{1}{2} \frac{2}{5} {theta}^{\frac{5}{2} }$

=$=0.2(\frac{3\pi}{2} )^5\\=1.519\pi$