14–25. {Use of Tech} Areas of regions Determine the area of the given region.


The region bounded by y = ln x,y = 1, and x = 1

Verified step by step guidance

1. Step 1: Understand the problem. The goal is to find the area of the region bounded by the curves y = ln(x), y = 1, and x = 1. This involves setting up an integral to calculate the area between these boundaries.
2. Step 2: Identify the limits of integration. The region is bounded by x = 1 and the vertical line where y = 1 intersects y = ln(x). To find this intersection, solve ln(x) = 1. This gives x = e (Euler's number). Thus, the limits of integration are from x = 1 to x = e.
3. Step 3: Set up the integral. The area is calculated by integrating the difference between the upper curve (y = 1) and the lower curve (y = ln(x)) over the interval [1, e]. The integral is: $\int x_1^e(1-\ln (x\left)\right)dx$
4. Step 4: Break down the integral. The integral can be split into two parts: $\int x_1^{e}1dx$ and $-\int x_1^e\ln \left(x\right)dx$. Evaluate each part separately.
5. Step 5: Evaluate the integral. For the first part, $\int x_1^{e}1dx$, the result is simply the length of the interval, e - 1. For the second part, $-\int x_1^e\ln \left(x\right)dx$, use integration by parts where u = ln(x) and dv = dx. Combine the results to find the total area.

Key Concepts

Definite Integrals

Definite integrals are used to calculate the area under a curve between two points on the x-axis. In this context, the area of the region bounded by the curves can be found by integrating the function y = ln(x) from the lower limit x = 1 to the upper limit where y = 1 intersects the curve.

Natural Logarithm Function

The natural logarithm function, denoted as ln(x), is the inverse of the exponential function e^x. It is defined for x > 0 and is crucial in this problem as it describes one of the boundaries of the region whose area we need to calculate.

Area Between Curves

The area between curves is determined by finding the difference between the upper and lower functions over a specified interval. In this case, the area is calculated by integrating the difference between y = 1 and y = ln(x) from x = 1 to the point where ln(x) equals 1.