evaluate the definite integral  $$ \int_0^3\!4x^3\text{ d}x $$

To evaluate the definite integral $ \int_0^3 4x^3 \, \text{d}x $, follow these steps:

### Step 1: Find the Indefinite Integral
First, find the indefinite integral (antiderivative) of the integrand $ 4x^3 $:
\[
\int 4x^3 \, \text{d}x
\]
Using the power rule for integration, which states:
\[
\int x^n \, \text{d}x = \frac{x^{n+1}}{n+1} + C
\]
where $ C $ is the constant of integration, we get:
\[
\int 4x^3 \, \text{d}x = 4 \cdot \frac{x^{3+1}}{3+1} = 4 \cdot \frac{x^4}{4} = x^4
\]

### Step 2: Apply the Limits of Integration
Now, evaluate the definite integral by applying the limits from 0 to 3:
\[
\int_0^3 4x^3 \, \text{d}x = \left[ x^4 \right]_0^3
\]
This means we calculate:
\[
\left[ x^4 \right]_0^3 = (3^4) - (0^4)
\]

### Step 3: Simplify the Result
Calculate the values:
\[
3^4 = 81 \quad \text{and} \quad 0^4 = 0
\]
Thus:
\[
\int_0^3 4x^3 \, \text{d}x = 81 - 0 = 81
\]

### Final Answer:
\[
\int_0^3 4x^3 \, \text{d}x = 81
\]

Would you like more details or have any questions?

Here are 5 related questions for further exploration:
1. How would the result change if the integrand were $ 4x^2 $ instead of $ 4x^3 $?
2. Can you evaluate the definite integral $ \int_1^4 x^2 \, \text{d}x $?
3. How do you handle definite integrals where the limits are negative?
4. What is the geometric interpretation of a definite integral?
5. How does the Fundamental Theorem of Calculus connect differentiation and integration?

**Tip:** Always remember to apply the limits of integration after finding the indefinite integral to evaluate a definite integral.

Learn how to evaluate the definite integral ∫ 4x^3 dx from 0 to 3 step-by-step. This problem involves using the power rule for integration and applying the Fundamental Theorem of Calculus. Suitable for advanced high school students and calculus learners.

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