To solve the given problem, we will convert the given limit expression into a definite integral and then evaluate that integral over the specified interval from [tex]\(2\)[/tex] to [tex]\(8\)[/tex].
The limit expression given is:
[tex]\[ \lim _{n \rightarrow \infty} \sum_{i=1}^n\left[4\left(x_i^\right)^3-3 x_i^\right] \Delta x \][/tex]
Here, this limit expression represents the Riemann sum of a function over the interval [tex]\([2, 8]\)[/tex].
1. Identifying the Function:
The function to be integrated is given by the terms inside the summation:
[tex]\[ f(x) = 4x^3 - 3x \][/tex]
2. Setting Up the Definite Integral:
The limit of the Riemann sum as [tex]\( n \to \infty \)[/tex] is equivalent to the definite integral of the function [tex]\( f(x) \)[/tex] over the interval [tex]\([2, 8]\)[/tex]. This can be expressed as:
[tex]\[ \int_{2}^{8} (4x^3 - 3x) \, dx \][/tex]
3. Evaluating the Definite Integral:
To evaluate the definite integral, we need to find the antiderivative of [tex]\( 4x^3 - 3x \)[/tex]. Let's find the antiderivative of each term separately:
- The antiderivative of [tex]\( 4x^3 \)[/tex] is:
[tex]\[ \int 4x^3 \, dx = x^4 \][/tex]
- The antiderivative of [tex]\( -3x \)[/tex] is:
[tex]\[ \int -3x \, dx = -\frac{3}{2} x^2 \][/tex]
Therefore, the antiderivative of [tex]\( 4x^3 - 3x \)[/tex] is:
[tex]\[ \int (4x^3 - 3x) \, dx = x^4 - \frac{3}{2} x^2 \][/tex]
4. Applying the Limits of Integration:
Now we need to evaluate this antiderivative from [tex]\( x = 2 \)[/tex] to [tex]\( x = 8 \)[/tex]:
[tex]\[
\left. \left( x^4 - \frac{3}{2} x^2 \right) \right|_2^8 = \left( 8^4 - \frac{3}{2} (8^2) \right) - \left( 2^4 - \frac{3}{2} (2^2) \right)
\][/tex]
5. Simplifying and Calculating:
- Calculate [tex]\( 8^4 \)[/tex] and [tex]\( 8^2 \)[/tex]:
[tex]\[ 8^4 = 4096 \][/tex]
[tex]\[ 8^2 = 64 \][/tex]
[tex]\[ \frac{3}{2} (8^2) = \frac{3}{2} \cdot 64 = 96 \][/tex]
- Calculate [tex]\( 2^4 \)[/tex] and [tex]\( 2^2 \)[/tex]:
[tex]\[ 2^4 = 16 \][/tex]
[tex]\[ 2^2 = 4 \][/tex]
[tex]\[ \frac{3}{2} (2^2) = \frac{3}{2} \cdot 4 = 6 \][/tex]
- Substitute these results back into the antiderivative expression:
[tex]\[
\left( 4096 - 96 \right) - \left( 16 - 6 \right) = 4000 - 10 = 3990
\][/tex]
Thus, the value of the definite integral is:
[tex]\[
\int_{2}^{8} (4x^3 - 3x) \, dx = 3990
\][/tex]
Therefore, the final result is:
[tex]\[
3990
\][/tex]
Hence, the definite integral of the given function over the interval [tex]\([2, 8]\)[/tex] is [tex]\(3990\)[/tex].
