Find the area of the region under the curve y = 16 e ^{4 x} between x = -1.4 to x =1.4 .
To find the area under the curve y = 16e^(4x) between x = -1.4 and x = 1.4, you can use definite integration. The definite integral of a function over a specified interval gives you the area under the curve between those two points.
Here's how you can calculate it step by step:
Step 1: Determine the integral expression:
∫[from -1.4 to 1.4] 16e^(4x) dx
Step 2: Solve the integral:
∫[from -1.4 to 1.4] 16e^(4x) dx
= [4e^(4x)] [from -1.4 to 1.4]
= 4e^(4 * 1.4) - 4e^(4 * -1.4)
Step 3: Calculate the values:
Now we need to calculate the exponential terms:
e^(4 * 1.4) ≈ 140.60
e^(4 * -1.4) ≈ 0.0071
Step 4: Substitute the values back into the integral expression:
4e^(4 * 1.4) - 4e^(4 * -1.4)
= 4 * 140.60 - 4 * 0.0071
Step 5: Perform the arithmetic:
= 562.40 - 0.0284
= 562.3716
So, the area under the curve y = 16e^(4x) between x = -1.4 and x = 1.4 is approximately 562.3716 square units.