Find the area under the curve y = e^x from x = 0 to x = 1.

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Question: Find the area under the curve y = e^x from x = 0 to x = 1.

Options:

Correct Answer: e - 1

Solution:

The area is given by the integral from 0 to 1 of e^x dx. This evaluates to [e^x] from 0 to 1 = e - 1.

Find the area under the curve y = e^x from x = 0 to x = 1.

Find the area under the curve y = e^x from x = 0 to x = 1.

Step 1: Identify the function you want to find the area under. In this case, it is y = e^x.
Step 2: Determine the limits for the area you want to calculate. Here, the limits are from x = 0 to x = 1.
Step 3: Set up the integral to find the area. This is written as the integral from 0 to 1 of e^x dx.
Step 4: Calculate the integral. The integral of e^x is e^x itself.
Step 5: Evaluate the integral at the upper limit (x = 1) and the lower limit (x = 0). This means you calculate e^1 - e^0.
Step 6: Simplify the result. e^1 is just e, and e^0 is 1, so you get e - 1.
Step 7: Conclude that the area under the curve from x = 0 to x = 1 is e - 1.

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