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 6.2.57
Chapter 6, Problem 6.2.57

Find the area of the region described in the following exercises.


The region bounded by y=x^2−2x+1 and y=5x−9

Verified step by step guidance
1
Step 1: Identify the curves that bound the region. The first curve is y = x^2 - 2x + 1, which is a parabola, and the second curve is y = 5x - 9, which is a straight line.
Step 2: Find the points of intersection between the two curves by setting their equations equal to each other: x^2 - 2x + 1 = 5x - 9. Rearrange the equation to form a standard quadratic equation: x^2 - 7x + 10 = 0.
Step 3: Solve the quadratic equation x^2 - 7x + 10 = 0 using factoring, the quadratic formula, or another method. The solutions will give the x-coordinates of the points of intersection.
Step 4: Set up the integral to calculate the area between the curves. The area is given by the integral of the difference between the upper curve (y = 5x - 9) and the lower curve (y = x^2 - 2x + 1) over the interval determined by the points of intersection. The integral is: ∫[a, b] ((5x - 9) - (x^2 - 2x + 1)) dx, where 'a' and 'b' are the x-coordinates of the intersection points.
Step 5: Simplify the integrand ((5x - 9) - (x^2 - 2x + 1)) to get -x^2 + 7x - 10. Then, compute the integral ∫[a, b] (-x^2 + 7x - 10) dx by finding the antiderivative and evaluating it at the bounds 'a' and 'b'.

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.

Finding the Area Between Curves

To find the area between two curves, we first identify the points of intersection, which serve as the limits of integration. The area is then calculated by integrating the difference between the upper curve and the lower curve over the interval defined by these points. This process involves setting up the integral of the function representing the upper curve minus the function representing the lower curve.
Recommended video:
05:23
Finding Area Between Curves on a Given Interval

Identifying Functions and Their Intersections

In this problem, we have two functions: a quadratic function, y = x^2 - 2x + 1, and a linear function, y = 5x - 9. To find the area between them, we need to determine where these two functions intersect. This is done by setting them equal to each other and solving for x, which gives us the points that will be used as limits for integration.
Recommended video:
6:13
Exponential Functions

Integration Techniques

Integration is a fundamental concept in calculus used to calculate areas under curves. In this context, we will use definite integration to find the area between the two curves. The integral of a function can be computed using various techniques, such as substitution or integration by parts, depending on the complexity of the functions involved.
Recommended video:
06:18
Integration by Parts for Definite Integrals
Related Practice
Textbook Question

60–63. Equivalent constant velocity Consider the following velocity functions. In each case, complete the sentence: The same distance could have been traveled over the given time period at a constant velocity of ________.


v(t)=2 sin t, for 0≤t≤π

Textbook Question

Find the area of the region described in the following exercises.


The region bounded by y=√x, y=2x−15, and y=0

Textbook Question

46–50. Force on dams The following figures show the shapes and dimensions of small dams. Assuming the water level is at the top of the dam, find the total force on the face of the dam.


Textbook Question

Find the area of the surface generated when the given curve is revolved about the given axis.


y=(3x)^1/3 , for 0≤x≤8/3; about the y-axis

Textbook Question

39–44. Shell method about other lines Let R be the region bounded by y = x²,x=1, and y=0. Use the shell method to find the volume of the solid generated when R is revolved about the following lines.


y = 2

Textbook Question

9-34. Shell method Let R be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when R is revolved about indicated axis. 


{Use of Tech} y = √50 -2x², in the first quadrant; about the x-axis