1. **State the problem:** Calculate the value of $$\sqrt{\int_{3\pi/8}^{7\pi/8} 40^2 \, dx + \frac{\int_{7\pi/8}^{2\pi+3\pi/8} (-80)^2 \, dx}{2\pi}}$$. 2. **Recall the integral formula:** For a constant $c$, $$\int_a^b c^2 \, dx = c^2 (b - a)$$. 3. **Calculate each integral:** - First integral: $$\int_{3\pi/8}^{7\pi/8} 40^2 \, dx = 1600 \times \left(\frac{7\pi}{8} - \frac{3\pi}{8}\right) = 1600 \times \frac{4\pi}{8} = 1600 \times \frac{\pi}{2} = 800\pi$$ - Second integral: $$\int_{7\pi/8}^{2\pi + 3\pi/8} (-80)^2 \, dx = 6400 \times \left(2\pi + \frac{3\pi}{8} - \frac{7\pi}{8}\right) = 6400 \times \left(2\pi - \frac{4\pi}{8}\right) = 6400 \times \left(2\pi - \frac{\pi}{2}\right) = 6400 \times \frac{3\pi}{2} = 9600\pi$$ 4. **Substitute into the expression:** $$\sqrt{800\pi + \frac{9600\pi}{2\pi}} = \sqrt{800\pi + \frac{9600\pi}{2\pi}}$$ Simplify the fraction inside the square root: $$\frac{9600\pi}{2\pi} = \frac{9600}{2} = 4800$$ So the expression becomes: $$\sqrt{800\pi + 4800}$$ 5. **Final answer:** $$\boxed{\sqrt{800\pi + 4800}}$$ This is the exact simplified form of the expression.