Skip to main content
Ch. 6 - Applications of Integration
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 6 - Applications of IntegrationProblem 6.R.17
Chapter 6, Problem 6.R.17

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.
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.
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: ∫x1e(1-ln(x))dx
Step 4: Break down the integral. The integral can be split into two parts: ∫x1e1dx and -∫x1eln(x)dx. Evaluate each part separately.
Step 5: Evaluate the integral. For the first part, ∫x1e1dx, the result is simply the length of the interval, e - 1. For the second part, -∫x1eln(x)dx, use integration by parts where u = ln(x) and dv = dx. Combine the results to find the total area.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
5m
Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.
Recommended video:
05:43
Definition of the Definite Integral

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.
Recommended video:
05:18
Derivative of the Natural Logarithmic Function

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.
Recommended video:
05:23
Finding Area Between Curves on a Given Interval
Related Practice
Textbook Question

{Use of Tech} Decreasing velocity A projectile is fired upward, and its velocity (in m/s) is given by v(t) = 200 / √t+1, for t≥0.

a. Graph the velocity function, for t≥0.

Textbook Question

Surface area and volume Let f(x) = 1/3 x³ and let R be the region bounded by the graph of f and the x-axis on the interval [0, 2].


c. Find the volume of the solid generated when R is revolved about the x-axis.

Textbook Question

43–55. Volumes of solids Choose the general slicing method, the disk/washer method, or the shell method to answer the following questions.


The region bounded by the curves y = sec x and y=2, for 0 ≤ x ≤ π/3, is revolved about the x-axis. What is the volume of the solid that is generated? 

1
views
Textbook Question

43–55. Volumes of solids Choose the general slicing method, the disk/washer method, or the shell method to answer the following questions.


What is the volume of the solid whose base is the region in the first quadrant bounded by y = √x,y = 2-x, and the x-axis, and whose cross sections perpendicular to the base and parallel to the y-axis are semicircles?

1
views
Textbook Question

82–84. Fluid Forces Suppose the following plates are placed on a vertical wall so that the top of the plate is 2 m below the surface of a pool that is filled with water. Compute the force on each plate.


A circular plate with a radius of 2 m

1
views
Textbook Question

27–33. Multiple regions The regions R₁,R₂, and R₃ (see figure) are formed by the graphs of y = 2√x,y = 3−x,and x=3.


Find the volume of the solid obtained by revolving region R₂ about the y-axis.

2
views