Find the area between the curves.
\[
x=-1, x=3, y=4 e^{4 x}, y=3 e^{4 x}+1
\]
The area between the curves is approximately
(Do not round until the final answer. Then round to the nearest hundredth as needed.)
Answer
Final Answer: The area between the curves is approximately \(\boxed{40686}\).
Step 1 ：The area between two curves is given by the integral of the absolute difference of the two functions over the given interval. In this case, the two functions are \(y=4e^{4x}\) and \(y=3e^{4x}+1\). The interval is from \(x=-1\) to \(x=3\).
Step 2 ：We need to find the integral of \(|4e^{4x} - (3e^{4x}+1)|\) from \(x=-1\) to \(x=3\).
Step 3 ：By evaluating the integral, we find that the area between the curves is approximately 40686.
Step 4 ：Final Answer: The area between the curves is approximately \(\boxed{40686}\).
