Determine the area under the curve y = 1/x from x = 1 to x = 2.

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Question: Determine the area under the curve y = 1/x from x = 1 to x = 2.

Options:

Correct Answer: ln(2)

Solution:

The area under the curve y = 1/x from x = 1 to x = 2 is given by ∫(from 1 to 2) (1/x) dx = [ln(x)] from 1 to 2 = ln(2) - ln(1) = ln(2).

Determine the area under the curve y = 1/x from x = 1 to x = 2.

Determine the area under the curve y = 1/x from x = 1 to x = 2.

Correct Answer: ln(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: Find the antiderivative of 1/x, which is ln(x).
Step 5: Evaluate the antiderivative at the upper limit (x = 2) and the lower limit (x = 1): ln(2) - ln(1).
Step 6: Simplify the result. Since ln(1) = 0, the area is ln(2) - 0 = ln(2).

Definite Integral – The question tests the understanding of calculating the area under a curve using definite integrals.
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