https://file.aiquickdraw.com/mathbot/file/1725526135944inj61a5g.jpg

I have opened the image you uploaded. Let me analyze the mathematical content and provide detailed solutions for the problems in the image.

It seems there are multiple questions, including one related to calculating the area enclosed by a graph of a function. I'll begin with the analysis and solving.

Please give me a moment to solve the problems.### Problem Breakdown:

From the image, the visible questions are as follows:

1. **Derive a solution using differentiation**:
   It involves finding the derivative of a function, though it's not fully clear. There seems to be a problem involving a trigonometric function and its derivative.

2. **Determine the integral**:
   The second visible question seems to ask for the integration of the function \( 6x^2 - x^3 \).

3. **Calculate the area enclosed by the curve**:
   This involves finding the area under a parabolic curve using integration. The limits of integration are given, as the problem is asking for the area between \( x = 0 \) and \( x = 4 \).

I'll start by solving the integral problem related to the area of the curve:

\[
f(x) = 6x^2 - x^3
\]
The limits are \( x = 0 \) to \( x = 4 \).

#### Step 1: Find the definite integral of the function \( f(x) = 6x^2 - x^3 \).

Let me calculate that now.

Calculate the area enclosed by the curve y = 6x^2 - x^3 from x = 0 to x = 4, using integration.

This problem demonstrates how to calculate the area under the curve y = 6x^2 - x^3 using definite integrals over the interval from x = 0 to x = 4. Step-by-step instructions include the use of calculus techniques and the application of the Fundamental Theorem of Calculus.

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