1. **State the problem:** Evaluate the integral $$\int_0^{\frac{3\pi}{2}} \sqrt{1 - \sin^2 t} \, dt$$. 2. **Recall the identity:** We know that $$\sin^2 t + \cos^2 t = 1$$, so $$1 - \sin^2 t = \cos^2 t$$. 3. **Simplify the integrand:** $$\sqrt{1 - \sin^2 t} = \sqrt{\cos^2 t} = |\cos t|$$. 4. **Rewrite the integral:** $$\int_0^{\frac{3\pi}{2}} |\cos t| \, dt$$. 5. **Analyze the behavior of $\cos t$ on $[0, \frac{3\pi}{2}]$:** - On $[0, \frac{\pi}{2}]$, $\cos t \geq 0$, so $|\cos t| = \cos t$. - On $[\frac{\pi}{2}, \frac{3\pi}{2}]$, $\cos t \leq 0$, so $|\cos t| = -\cos t$. 6. **Split the integral accordingly:** $$\int_0^{\frac{3\pi}{2}} |\cos t| \, dt = \int_0^{\frac{\pi}{2}} \cos t \, dt + \int_{\frac{\pi}{2}}^{\frac{3\pi}{2}} -\cos t \, dt$$. 7. **Evaluate each integral:** - $$\int_0^{\frac{\pi}{2}} \cos t \, dt = \sin t \Big|_0^{\frac{\pi}{2}} = 1 - 0 = 1$$. - $$\int_{\frac{\pi}{2}}^{\frac{3\pi}{2}} -\cos t \, dt = -\sin t \Big|_{\frac{\pi}{2}}^{\frac{3\pi}{2}} = - (\sin \frac{3\pi}{2} - \sin \frac{\pi}{2}) = - (-1 - 1) = 2$$. 8. **Sum the results:** $$1 + 2 = 3$$. **Final answer:** $$\int_0^{\frac{3\pi}{2}} \sqrt{1 - \sin^2 t} \, dt = 3$$.