To evaluate the definite integral of the function [tex]\( t^2 - 1 \)[/tex] from [tex]\(-5\)[/tex] to [tex]\(5\)[/tex], we will follow these steps:
1. Set Up the Integral:
We need to evaluate the integral:
[tex]\[
\int_{-5}^{5} (t^2 - 1) \, dt
\][/tex]
2. Find the Antiderivative:
The antiderivative of [tex]\( t^2 \)[/tex] is [tex]\(\frac{t^3}{3}\)[/tex], and the antiderivative of [tex]\(-1\)[/tex] is [tex]\(-t\)[/tex]. Therefore, the antiderivative of [tex]\( t^2 - 1 \)[/tex] is:
[tex]\[
F(t) = \frac{t^3}{3} - t
\][/tex]
3. Evaluate the Antiderivative at the Limits of Integration:
We evaluate [tex]\( F(t) \)[/tex] at the upper limit 5 and the lower limit -5:
[tex]\[
F(5) = \frac{5^3}{3} - 5 = \frac{125}{3} - 5
\][/tex]
Simplifying, we get:
[tex]\[
F(5) = \frac{125}{3} - \frac{15}{3} = \frac{110}{3}
\][/tex]
Similarly, at the lower limit -5:
[tex]\[
F(-5) = \frac{(-5)^3}{3} - (-5) = \frac{-125}{3} + 5
\][/tex]
Simplifying, we get:
[tex]\[
F(-5) = \frac{-125}{3} + \frac{15}{3} = \frac{-110}{3}
\][/tex]
4. Compute the Definite Integral:
We now subtract the value of the antiderivative at the lower limit from the value at the upper limit:
[tex]\[
\int_{-5}^{5} (t^2 - 1) \, dt = F(5) - F(-5) = \frac{110}{3} - \left( \frac{-110}{3} \right)
\][/tex]
Simplifying the expression inside the parentheses, we get:
[tex]\[
\int_{-5}^{5} (t^2 - 1) \, dt = \frac{110}{3} + \frac{110}{3} = \frac{220}{3}
\][/tex]
Therefore, the value of the definite integral is:
[tex]\[
\int_{-5}^{5} (t^2 - 1) \, dt = \frac{220}{3} \approx 73.33333333333334
\][/tex]
Verification Using a Graphing Utility:
To verify the result, you can use a graphing utility such as a graphing calculator or software that allows you to input and evaluate definite integrals. Input the function [tex]\( t^2 - 1 \)[/tex] and the limits of integration from [tex]\(-5\)[/tex] to [tex]\(5\)[/tex], and you should observe that the calculated area under the curve matches our result, approximately 73.33333333333334.
