To find the integral of the absolute value of $\sin(x)$ from $0$ to $\frac{3\pi}{2}$, we need to split the integral into intervals where $\sin(x)$ maintains a consistent sign, as the absolute value affects the function differently based on whether the sine function is positive or negative.

Step-by-Step Solution

1. Identify intervals where $\sin(x)$ is positive or negative between $0$ and $\frac{3\pi}{2}$:

   • From $0$ to $\pi$, $\sin(x) \geq 0$.
   • From $\pi$ to $\frac{3\pi}{2}$, $\sin(x) \leq 0$.

2. Set up the integrals over these intervals:

   • For $0 \leq x \leq \pi$, $|\sin(x)| = \sin(x)$.
   • For $\pi \leq x \leq \frac{3\pi}{2}$, $|\sin(x)| = -\sin(x)$.

   Therefore, we split the original integral as: $\int_0^{\frac{3\pi}{2}} |\sin(x)| \, dx = \int_0^{\pi} \sin(x) \, dx + \int_{\pi}^{\frac{3\pi}{2}} -\sin(x) \, dx$

3. Evaluate each integral:

   • For the first integral: $\int_0^{\pi} \sin(x) \, dx = \left[-\cos(x)\right]_0^{\pi} = -\cos(\pi) - (-\cos(0)) = -(-1) - (-1) = 2$

   • For the second integral: $\int_{\pi}^{\frac{3\pi}{2}} -\sin(x) \, dx = -\left[\cos(x)\right]_{\pi}^{\frac{3\pi}{2}} = -\left(\cos\left(\frac{3\pi}{2}\right) - \cos(\pi)\right)$ Since $\cos\left(\frac{3\pi}{2}\right) = 0$ and $\cos(\pi) = -1$, $-\left(0 - (-1)\right) = -1$

4. Add the results of the integrals: $\int_0^{\frac{3\pi}{2}} |\sin(x)| \, dx = 2 + 1 = 3$

Therefore, the integral of $|\sin(x)|$ from $0$ to $\frac{3\pi}{2}$ is $3$.

Would you like more details or have any questions about this solution?

Relative Questions

1. How would the integral change if the limits were from $0$ to $2\pi$?
2. What is the general strategy for integrating the absolute value of a trigonometric function?
3. How can we integrate other functions involving absolute values?
4. What role does symmetry play in integrals involving trigonometric functions?
5. How would this integral differ if it were $\int_0^{\frac{3\pi}{2}} |\cos(x)| \, dx$?

Tip

When integrating absolute values of trigonometric functions, always identify the intervals where the function is positive or negative to simplify the absolute value expressions.