To find the area enclosed between the curves $y = 3\sin(x)$ and $y = 2\cos(x)$ from $x = 0$ to $x = \frac{\pi}{2}$, we need to follow these steps:

Step 1: Set up the integral

The area between two curves from $x = a$ to $x = b$ is given by the integral of the difference of the functions, where the upper function is subtracted from the lower function:

$A = \int_a^b \left( f(x) - g(x) \right) dx$

In our case, $f(x) = 3\sin(x)$ and $g(x) = 2\cos(x)$, so the area can be written as:

$A = \int_0^{\frac{\pi}{2}} \left( 3\sin(x) - 2\cos(x) \right) dx$

Step 2: Compute the integral

We now integrate the expression:

$A = \int_0^{\frac{\pi}{2}} 3\sin(x) \, dx - \int_0^{\frac{\pi}{2}} 2\cos(x) \, dx$

We can compute each integral separately:

1. For $\int 3\sin(x) \, dx$, the antiderivative is $-3\cos(x)$.
2. For $\int 2\cos(x) \, dx$, the antiderivative is $2\sin(x)$.

Step 3: Evaluate the integrals

We evaluate the integrals from $x = 0$ to $x = \frac{\pi}{2}$:

1. $\left[ -3\cos(x) \right]_0^{\frac{\pi}{2}} = -3\left(\cos\left(\frac{\pi}{2}\right) - \cos(0)\right) = -3(0 - 1) = 3$
2. $\left[ 2\sin(x) \right]_0^{\frac{\pi}{2}} = 2\left(\sin\left(\frac{\pi}{2}\right) - \sin(0)\right) = 2(1 - 0) = 2$

Step 4: Final computation

Now subtract the results:

$A = 3 - 2 = 1$

Conclusion:

The area enclosed between $y = 3\sin(x)$ and $y = 2\cos(x)$ from $x = 0$ to $x = \frac{\pi}{2}$ is 1 square unit.

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Would you like more details on any step or have any questions about the process?

Related Questions:

1. How can you find the area between two curves when they intersect?
2. How do you handle situations where the functions are not simple trigonometric functions?
3. What happens if the integral gives a negative result? How do you interpret that in terms of area?
4. How can you generalize the process for finding areas enclosed between more than two curves?
5. What other methods exist for finding the area of irregular regions, such as numerical methods?

Tip:

When setting up integrals for areas between curves, always double-check which function is the upper function and which is the lower to ensure that you're subtracting correctly!