To solve the definite integral [tex]\(\int_4^3 3x^2 \, dx\)[/tex], we will follow these steps:
1. Find the indefinite integral:
The first step is to find the indefinite integral of the function [tex]\(3x^2\)[/tex].
[tex]\[
\int 3x^2 \, dx
\][/tex]
To integrate [tex]\(3x^2\)[/tex], we use the power rule for integration which states that [tex]\(\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\)[/tex].
Applying this to [tex]\(3x^2\)[/tex]:
[tex]\[
\int 3x^2 \, dx = 3 \int x^2 \, dx = 3 \left( \frac{x^3}{3} \right) + C = x^3 + C
\][/tex]
So, the indefinite integral of [tex]\(3x^2\)[/tex] is:
[tex]\[
x^3 + C
\][/tex]
2. Evaluate the definite integral:
Now, we need to evaluate the definite integral [tex]\(\int_4^3 3x^2 \, dx\)[/tex] by using the result of the indefinite integral. The definite integral is evaluated by finding the difference between the values of the antiderivative at the upper and lower limits of integration.
The definite integral [tex]\(\int_4^3 3x^2 \, dx\)[/tex] is given by:
[tex]\[
\left[ x^3 \right]_4^3
\][/tex]
This means we need to compute [tex]\(x^3\)[/tex] at the bounds 3 and 4, and then subtract the value at 4 from the value at 3.
[tex]\[
\left[ x^3 \right]_4^3 = x^3 \Bigg|_4 - x^3 \Bigg|_3
\][/tex]
Calculate [tex]\(x^3\)[/tex] at [tex]\(x = 3\)[/tex]:
[tex]\[
3^3 = 27
\][/tex]
Calculate [tex]\(x^3\)[/tex] at [tex]\(x = 4\)[/tex]:
[tex]\[
4^3 = 64
\][/tex]
3. Subtract the value at the lower limit from the value at the upper limit:
[tex]\[
27 - 64 = -37
\][/tex]
Therefore, the value of the definite integral [tex]\(\int_4^3 3x^2 \, dx\)[/tex] is [tex]\(-37\)[/tex].
