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

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

Options:

Correct Answer: e - 1

Solution:

The area under the curve y = e^x from 0 to 1 is given by ∫(from 0 to 1) e^x dx = [e^x] from 0 to 1 = e - 1.

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

Determine 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, the function is y = e^x.
Step 2: Determine the limits of integration. We want to find the area from x = 0 to x = 1.
Step 3: Set up the integral to calculate the area. This is written as ∫(from 0 to 1) e^x dx.
Step 4: Find the antiderivative of e^x. The antiderivative of e^x is e^x itself.
Step 5: Evaluate the antiderivative at the upper limit (x = 1) and the lower limit (x = 0). This means you calculate e^1 and e^0.
Step 6: Calculate e^1, which is e, and e^0, which is 1.
Step 7: Subtract the value at the lower limit from the value at the upper limit: e - 1.
Step 8: The result, e - 1, is the area under the curve from x = 0 to x = 1.

Definite Integral – The process of calculating the area under a curve between two specified points using integration.
Exponential Function – Understanding the properties and behavior of the exponential function, particularly e^x.