To evaluate the definite integral [tex]\(\int_2^4 x^2 \, dx\)[/tex], we can follow these steps:
1. Find the antiderivative of the integrand [tex]\(x^2\)[/tex]:
The integrand is [tex]\(x^2\)[/tex]. The antiderivative of [tex]\(x^2\)[/tex] is found by using the power rule for integration, which states:
[tex]\[
\int x^n \, dx = \frac{x^{n+1}}{n+1} + C,
\][/tex]
where [tex]\(n\)[/tex] is a constant and [tex]\(C\)[/tex] is the constant of integration. In this case, [tex]\(n = 2\)[/tex]:
[tex]\[
\int x^2 \, dx = \frac{x^{2+1}}{2+1} + C = \frac{x^3}{3} + C.
\][/tex]
2. Evaluate the definite integral from 2 to 4:
The definite integral from [tex]\(a\)[/tex] to [tex]\(b\)[/tex] of a function [tex]\(f(x)\)[/tex] is given by evaluating the antiderivative at the upper limit and subtracting the value of the antiderivative at the lower limit:
[tex]\[
\int_a^b f(x) \, dx = F(b) - F(a),
\][/tex]
where [tex]\(F(x)\)[/tex] is the antiderivative of [tex]\(f(x)\)[/tex].
Here, the antiderivative [tex]\(F(x)\)[/tex] is [tex]\(\frac{x^3}{3}\)[/tex]. Thus, we need to evaluate this at [tex]\(x = 4\)[/tex] and [tex]\(x = 2\)[/tex]:
[tex]\[
F(4) = \frac{4^3}{3} = \frac{64}{3},
\][/tex]
[tex]\[
F(2) = \frac{2^3}{3} = \frac{8}{3}.
\][/tex]
3. Subtract the values to find the definite integral:
[tex]\[
\int_2^4 x^2 \, dx = F(4) - F(2) = \frac{64}{3} - \frac{8}{3} = \frac{64 - 8}{3} = \frac{56}{3}.
\][/tex]
Therefore, the value of the definite integral [tex]\(\int_2^4 x^2 \, dx\)[/tex] is [tex]\(\frac{56}{3}\)[/tex].
