Find the points of intersection of the graphs of the functions y=5sin(6x)
 and y=0
 for x
 between x=0
 and x=3π12
 (enter your answer as a comma separated list) 0, pi/6
0, pi/6
0,π6
0, 
6
π
​
 
Then find the area bounded by the curves y=5sin(6x)
 and y=0
 for x
 between x=0
 and x=3π12

### 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

Find the points of intersection of the graphs of the functions y=5sin(6x) and y=0 for x between x=0 and x=3π/12. Then find the area bounded by the curves y=5sin(6x) and y=0 for x between x=0 and x=3π/12.

Explore the step-by-step solution to finding the points of intersection and the area bounded by the graphs of y=5sin(6x) and y=0 within the interval x=0 to x=3π/12. This problem involves key trigonometric and calculus concepts, making it ideal for high school students in Grades 10-12.

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