What is the area under the curve y = 1/x from x = 1 to x = 2?

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Question: What is the area under the curve y = 1/x from x = 1 to x = 2?

Options:

Correct Answer: ln(2)

Solution:

The area is given by the integral from 1 to 2 of 1/x dx. This evaluates to [ln(x)] from 1 to 2 = ln(2) - ln(1) = ln(2).

What is the area under the curve y = 1/x from x = 1 to x = 2?

What is the area under the curve y = 1/x from x = 1 to x = 2?

Step 1: Identify the function we are working with, which is y = 1/x.
Step 2: Determine the limits of integration, which are from x = 1 to x = 2.
Step 3: Set up the integral to find the area under the curve: ∫(from 1 to 2) (1/x) dx.
Step 4: Calculate the integral of 1/x, which is ln(x).
Step 5: Evaluate the integral from the lower limit (1) to the upper limit (2): [ln(x)] from 1 to 2.
Step 6: Substitute the limits into the evaluated integral: ln(2) - ln(1).
Step 7: Simplify the expression: ln(1) is 0, so the area is ln(2).

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