Textbook Question
Find the area of the region described in the following exercises.
The region bounded by y=|x−3|and y=x/2
Ch. 6 - Applications of Integration
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
Not the one you use?Change textbookSelect a different textbook
Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 6, Problem 6.2.13
Determine the area of the shaded region in the following figures.
(Hint: Find the intersection point by inspection.)
Verified step by step guidance1
Step 1: Identify the equations of the curves that bound the shaded region. The curves are y = 2^x (an exponential function) and y = 3 - x (a linear function).
Step 2: Find the intersection point of the two curves by solving the equation 2^x = 3 - x. This can be done by inspection or algebraically. The intersection point will provide the limits of integration.
Step 3: Set up the integral to calculate the area of the shaded region. The area is given by the integral of the upper curve minus the lower curve over the interval determined by the intersection points. The integral is: ∫[x1 to x2] ((3 - x) - (2^x)) dx, where x1 and x2 are the x-coordinates of the intersection points.
Step 4: Break down the integral into manageable parts. For example, split the integral into ∫[x1 to x2] (3 dx) - ∫[x1 to x2] (x dx) - ∫[x1 to x2] (2^x dx). Each term can be evaluated separately using standard integration techniques.
Step 5: Evaluate each integral. For ∫(3 dx), use the formula for the integral of a constant. For ∫(x dx), use the power rule. For ∫(2^x dx), use the formula for the integral of an exponential function. 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:
3mWas this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Intersection of Functions
The intersection of functions occurs where their graphs meet, meaning they have the same output for a given input. In this problem, finding the intersection points of the curves y = 2^x and y = 3 - x is essential, as these points define the boundaries of the shaded area whose area we need to calculate.
Recommended video:
02:59
Piecewise Functions Example 1
Area Under a Curve
The area under a curve can be calculated using definite integrals, which represent the accumulation of quantities. In this case, the area of the shaded region between the two curves can be found by integrating the difference between the upper function (3 - x) and the lower function (2^x) over the interval defined by their intersection points.
Recommended video:
05:59Estimating the Area Under a Curve with Right Endpoints & Midpoint
Definite Integral
A definite integral calculates the net area under a curve between two specified limits. It is denoted as ∫[a, b] f(x) dx, where f(x) is the function being integrated, and a and b are the limits of integration. This concept is crucial for determining the area of the shaded region in the given problem, as it provides a systematic way to compute the area between the two intersecting curves.
Recommended video:
05:43Definition of the Definite Integral
Related Practice
Textbook Question
What is the area of the curved surface of a right circular cone of radius 3 and height 4?
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.
y = x³−x⁸+1,y=1; about the y-axis
1
views
Textbook Question
53–62. Choose your method Let R be the region bounded by the following curves. Use the method of your choice to find the volume of the solid generated when R is revolved about the given axis.
y = x²,y=2−x, and x=0, in the first quadrant; about the y-axis
1
views
Textbook Question
Find the area of the region described in the following exercises.
The region bounded by y=2 / 1 + x^2 and y=1
Textbook Question
Determine the area of the shaded region bounded by the curve x^2=y^4(1−y^3) (see figure).
1
views