Find the area of the region between the graph of y=1/x, the x-axis, and the vertical lines x = 1 and x = 4.
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To find the area of the region between the graph of y = 1/x, the x-axis, and the vertical lines x = 1 and x = 4, we can use definite integrals.
First, we need to determine the limits of integration. In this case, we want to find the area between x = 1 and x = 4, so our limits of integration will be 1 and 4.
Next, we need to set up the integral. Since we are calculating the area, we can integrate the function f(x) = 1/x with respect to x over the given interval. However, we need to be careful about the range of the function, since it approaches infinity as x approaches 0. So, we need to split the integral into two parts: one from 1 to a small positive number close to 0, and another from this small positive number to 4.
The integral can be set up as follows:
∫[from 1 to 4] (1/x) dx = ∫[from 1 to ε] (1/x) dx + ∫[from ε to 4] (1/x) dx
where ε is the small positive number close to 0.
To calculate the integral, we can use the natural logarithm function. Recall that the integral of 1/x is ln(x), so the integrals can be simplified as follows:
= [ln|x|] from 1 to ε + [ln|x|] from ε to 4
Finally, we evaluate the integral:
= ln|ε| - ln|1| + ln|4| - ln|ε|
The ln|ε| terms will cancel each other out, leaving us with:
= ln|4| - ln|1| = ln(4)
Therefore, the area of the region between the graph of y = 1/x, the x-axis, and the vertical lines x = 1 and x = 4 is ln(4).