Let's solve the given problems step by step.

Soal 1

Given: The curve $y = x^3 - 3x^2 + 4$ and the x-axis, over the interval $x \in [1, 4]$.
Find: The area between the curve and the x-axis.

Solution

The formula to compute the area between a curve $y = f(x)$ and the x-axis is:

$\text{Area} = \int_{a}^{b} |f(x)| \, dx$

Here, $f(x) = x^3 - 3x^2 + 4$, and the interval is $[1, 4]$. First, we check where the curve crosses the x-axis by solving $f(x) = 0$:

$x^3 - 3x^2 + 4 = 0$

This can be solved to determine points of sign changes. Then, integrate separately where $f(x)$ is positive or negative.

Soal 2

Given: $v(t) = t^2 - 4t + 3$ is the velocity of a particle over $t \in [0, 5]$.

Part (a): Total Displacement

The total displacement is the integral of the velocity function:

$\text{Displacement} = \int_0^5 v(t) \, dt$

Substitute $v(t) = t^2 - 4t + 3$, and evaluate.

Part (b): Total Distance

To find the total distance, we integrate the absolute value of the velocity:

$\text{Distance} = \int_0^5 |v(t)| \, dt$

Identify when $v(t) = 0$ to check where the velocity changes sign. Solve:

$t^2 - 4t + 3 = 0 \implies (t-3)(t-1) = 0 \implies t = 1, t = 3$

Thus, split the integral into intervals $[0,1]$, $[1,3]$, and $[3,5]$, adjusting for the sign of $v(t)$ in each interval.

Soal 3

Given: $y = \ln\left( \sqrt{x^2 + e^{\sin(3x)}} \right) \cdot \cos^4\left( \frac{1}{x} \right)$.
Find: $\frac{dy}{dx}$.

Solution

To differentiate $y$, we use:

1. Product Rule: $(u \cdot v)' = u'v + uv'$
2. Chain Rule for composite functions.
3. Trigonometric Derivatives: Differentiating $\cos^n(u)$ and related functions.

Let $u = \ln\left( \sqrt{x^2 + e^{\sin(3x)}} \right)$ and $v = \cos^4\left( \frac{1}{x} \right)$.

• Differentiate $u$ using the chain rule:
  $u = \ln\left( \sqrt{x^2 + e^{\sin(3x)}} \right)$

• Differentiate $v$ using the chain and power rule.

Combine both results using the product rule to find $\frac{dy}{dx}$.

Would you like step-by-step calculations for any of these problems? Let me know!

5 Related Questions:

1. How do you calculate areas under a curve if the function crosses the x-axis?
2. What is the difference between displacement and distance in kinematics?
3. How do you solve integrals involving absolute values?
4. How do you use the product rule for differentiation?
5. How do you apply the chain rule to logarithmic and trigonometric functions?

Tip: When solving for total distance, always split the integral at points where the velocity changes sign to ensure you account for magnitude correctly.