Evaluate the definite integral. Use a graphing utility to verify your result.
\[
\int_{-1}^{1}\left(t^{2}-5\right) d t
\]
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Answer
Final Answer: \(\boxed{-\frac{28}{3}}\)
Step 1 ：First, we need to find the antiderivative of \(t^2 - 5\). The antiderivative of \(t^2\) is \(\frac{1}{3}t^3\) and the antiderivative of -5 is -5t. So, the antiderivative of \(t^2 - 5\) is \(\frac{1}{3}t^3 - 5t\).
Step 2 ：Next, we need to evaluate this antiderivative at 1 and -1, and subtract the two results. This will give us the value of the definite integral.
Step 3 ：The result of the definite integral from -1 to 1 of the function \(t^2 - 5\) is \(-\frac{28}{3}\). This is the area under the curve of the function from -1 to 1.
Step 4 ：Final Answer: \(\boxed{-\frac{28}{3}}\)
