Evaluate the integral and interpret it as the area of a region. ∫(from 0 to π/2) |5 sin(x) − 5 cos(2x)| dx

Steps to Solve

1. Identify the Function and Interval We have the function $f(x) = 5 \sin(x) - 5 \cos(2x)$ and we need to evaluate the integral from $0$ to $\frac{\pi}{2}$.

2. Simplify the Function Let's simplify the expression for $f(x)$: $$ f(x) = 5 \sin(x) - 5 \cos(2x) = 5 \sin(x) - 5(2 \sin^2(x) - 1) $$ $$ = 5 \sin(x) - 10 \sin^2(x) + 5 $$ Combine like terms: $$ f(x) = 5 + 5 \sin(x) - 10 \sin^2(x) $$

3. Determine the Sign of the Function Next, we want to find where the function is positive or negative on the interval $[0, \frac{\pi}{2}]$. Set $f(x) = 0$ to find critical points: $$ 5 \sin(x) - 10 \sin^2(x) + 5 = 0 $$ $$ 10 \sin^2(x) - 5 \sin(x) - 5 = 0 $$ Use the quadratic formula: $$ \sin(x) = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ Here, $a = 10$, $b = -5$, $c = -5$.

4. Calculate the Quadratic Roots Calculate the discriminant: $$ D = (-5)^2 - 4(10)(-5) = 25 + 200 = 225 $$ Now calculate the roots: $$ \sin(x) = \frac{5 \pm 15}{20} $$ Thus, we have:

5. $\sin(x) = 1$ (at $x = \frac{\pi}{2}$)

6. $\sin(x) = -\frac{1}{2}$ (not in our interval)

Since $f(0) = 5$, and $f(x)$ is continuous, we can conclude that $f(x) \geq 0$ over $[0, \frac{\pi}{2}]$.

5. Set Up the Integral Now that we have determined $f(x) \geq 0$, we can set up the integral: $$ \int_0^{\frac{\pi}{2}} f(x) , dx = \int_0^{\frac{\pi}{2}} (5 \sin(x) - 5 \cos(2x)) , dx $$

6. Evaluate the Integral We can break the integral into two parts: $$ \int_0^{\frac{\pi}{2}} f(x) , dx = 5 \int_0^{\frac{\pi}{2}} \sin(x) , dx - 5 \int_0^{\frac{\pi}{2}} \cos(2x) , dx $$

Calculating the first integral: $$ \int \sin(x) , dx = -\cos(x) + C $$ Thus, $$ 5 \int_0^{\frac{\pi}{2}} \sin(x) , dx = 5[-\cos(x)]_0^{\frac{\pi}{2}} = 5[0 - (-1)] = 5 $$

For the second integral: $$ \int \cos(2x) , dx = \frac{1}{2} \sin(2x) + C $$ Thus, $$ 5 \int_0^{\frac{\pi}{2}} \cos(2x) , dx = 5 \left[\frac{1}{2} \sin(2x)\right]_0^{\frac{\pi}{2}} = 5 \left[\frac{1}{2}(0) - \frac{1}{2}(0)\right] = 0 $$

7. Final Calculation Now combine the results: $$ \int_0^{\frac{\pi}{2}} f(x) , dx = 5 - 0 = 5 $$