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 6.2.55
Chapter 6, Problem 6.2.55

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


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

Verified step by step guidance
1
Step 1: Identify the boundaries of the region. The region is bounded by three curves: y = √x, y = 2x − 15, and y = 0. The curve y = 0 represents the x-axis.
Step 2: Determine the points of intersection between the curves y = √x and y = 2x − 15. To find these points, set √x = 2x − 15 and solve for x. This will give the x-coordinates where the two curves intersect.
Step 3: Establish the limits of integration. The region is bounded by the x-values where the curves intersect and where they meet the x-axis (y = 0). Solve for x when y = 0 for each curve to find these limits.
Step 4: Set up the integral to calculate the area. Divide the region into subregions if necessary, based on which curve is above the other. For each subregion, subtract the lower curve from the upper curve and integrate with respect to x over the appropriate interval.
Step 5: Evaluate the integral(s). Perform the integration for each subregion and sum the results to find the total area of the region. Ensure proper handling of the limits of integration and the subtraction of curves.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
6m
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 difference between the upper function and the lower function over the interval defined by their points of intersection.
Recommended video:
05:43
Definition of the Definite Integral

Finding Points of Intersection

To determine the area between the curves, it is essential to find the points where the curves intersect. This involves solving the equations of the functions simultaneously, which provides the limits of integration necessary for calculating the area.
Recommended video:
04:50
Critical Points

Area Between Curves

The area between two curves can be calculated by integrating the difference of the two functions over the interval defined by their intersection points. This method allows us to find the total area enclosed by the curves and the x-axis, which is crucial for solving the given problem.
Recommended video:
05:23
Finding Area Between Curves on a Given Interval
Related Practice
Textbook Question

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


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

Textbook Question

Let R be the region bounded by the following curves. Find the volume of the solid generated when R is revolved about the given axis.


y=4−x^2,x=2, and y=4; about the y-axis

Textbook Question

35–38. Shell and washer methods Let R be the region bounded by the following curves. Use both the shell method and the washer method to find the volume of the solid generated when R is revolved about the indicated axis.


y = (x−2)³ −2,x=0, and y=25; about the y-axis

Textbook Question

9–20. Arc length calculations Find the arc length of the following curves on the given interval.

y = x^3/2 / 3 − x^1/2 on [4, 16]

1
views
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