Step 1: Finding the Points of Intersection

We are tasked with finding the points of intersection between the graphs of $y = 5\sin(6x)$ and $y = 0$ in the interval $x \in [0, \frac{3\pi}{12}]$.

To do this, set the two equations equal to each other: $5\sin(6x) = 0$

This simplifies to: $\sin(6x) = 0$

The general solution for $\sin(6x) = 0$ is: $6x = n\pi \quad \text{for} \quad n \in \mathbb{Z}$ $x = \frac{n\pi}{6}$

Now, we need to find the values of $x$ in the interval $\left[0, \frac{3\pi}{12} \right]$, which simplifies to $\left[0, \frac{\pi}{4}\right]$.

For $n = 0$, we get $x = 0$.

For $n = 1$, we get: $x = \frac{\pi}{6}$

There are no more values of $x$ in the given interval.

Thus, the points of intersection are: $x = 0, \frac{\pi}{6}$

Step 2: Finding the Area Bounded by the Curves

We now find the area between $y = 5\sin(6x)$ and $y = 0$ in the interval $x \in [0, \frac{3\pi}{12}]$, i.e., $x \in [0, \frac{\pi}{4}]$.

The area under the curve is given by the integral: $\text{Area} = \int_0^{\frac{\pi}{6}} 5\sin(6x) \, dx$

To solve this integral: $\int 5\sin(6x) \, dx = -\frac{5}{6}\cos(6x)$

Now, evaluate this from $x = 0$ to $x = \frac{\pi}{6}$: $\text{Area} = \left[-\frac{5}{6}\cos(6x)\right]_0^{\frac{\pi}{6}}$ $= -\frac{5}{6} \left(\cos(6 \cdot \frac{\pi}{6}) - \cos(0)\right)$ $= -\frac{5}{6} \left(\cos(\pi) - \cos(0)\right)$ $= -\frac{5}{6} \left(-1 - 1\right)$ $= -\frac{5}{6} \cdot (-2) = \frac{10}{6} = \frac{5}{3}$

Thus, the area bounded by the curves is $\frac{5}{3}$.

Final Answers:

• Points of intersection: $0, \frac{\pi}{6}$
• Area bounded by the curves: $\frac{5}{3}$

Do you want further details on any step or have additional questions?

Here are 5 related questions to explore:

1. How would the area change if the function were $y = 10\sin(6x)$?
2. What are the solutions to $\sin(6x) = 0$ for $x$ in $[0, \pi]$?
3. How do you find the length of the curve $y = 5\sin(6x)$ between two points?
4. What is the maximum value of $y = 5\sin(6x)$ and at which $x$ does it occur?
5. How would you compute the area between two sinusoidal functions, like $y = 5\sin(6x)$ and ( y