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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 6.2.5a
Chapter 6, Problem 6.2.5a

Find the area of the region (see figure) in two ways.
a. Using integration with respect to x.
Graph showing the area between the curves y=2-x and y=x, with shaded region and labeled points (0,2) and (1,1).

Verified step by step guidance
1
Identify the curves and the region bounded by them. The region is bounded by the lines \(y = 2 - x\) and \(y = x\), between the points where they intersect.
Find the points of intersection by setting the two equations equal: \(2 - x = x\). Solve for \(x\) to find the intersection point(s).
Set up the integral with respect to \(x\). The area between the curves from the left intersection point to the right intersection point is given by the integral of the top function minus the bottom function: \(\int_{a}^{b} [(2 - x) - x] \, dx\).
Simplify the integrand to \(2 - 2x\) and write the definite integral with the limits found in step 2.
Evaluate the integral (without calculating the final value here) to find the area of the region.

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Key Concepts

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

Area Between Curves

The area between two curves is found by integrating the difference of the functions over the interval where they intersect. Specifically, if y = f(x) is above y = g(x), the area is the integral of (f(x) - g(x)) dx between the intersection points.
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Finding Area Between Curves on a Given Interval

Finding Points of Intersection

To determine the limits of integration, find where the two curves intersect by setting their equations equal and solving for x. These points define the interval over which the area is calculated.
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Critical Points

Integration with Respect to x

Integration with respect to x involves summing vertical slices of the region. Each slice has height equal to the difference between the upper and lower functions at a given x, and width dx, allowing calculation of the total area.
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Integrals of Natural Exponential Functions (e^x)
Related Practice
Textbook Question

40–43. Population growth


A culture of bacteria in a Petri dish has an initial population of 1500 cells and grows at a rate (in cells/day) of N′(t) = 100e^−0.25t. Assume t is measured in days.


a. What is the population after 20 days? After 40 days?

Textbook Question

Winding a chain A 30-m-long chain hangs vertically from a cylinder attached to a winch. Assume there is no friction in the system and the chain has a density of 5kg/m.

a. How much work is required to wind the entire chain onto the cylinder using the winch?

Textbook Question

Blood flow A typical human heart pumps 70 mL of blood (the stroke volume) with each beat. Assuming a heart rate of 60 beats/min (1 beat/s), a reasonable model for the outflow rate of the heart is V′(t)=70(1+sin 2πt), where V(t) is the amount of blood (in milliliters) pumped over the interval [0,t],V(0)=0 and t is measured in seconds.


a. Verify that the amount of blood pumped over a one-second interval is 70 mL.

Textbook Question

Region R is revolved about the line y=1 to form a solid of revolution.


a. What is the radius of a cross section of the solid at a point x in [0, 4]?

Textbook Question

6–8. Let R be the region bounded by the curves y = 2−√x,y=2, and x=4 in the first quadrant.


Suppose the shell method is used to determine the volume of the solid generated by revolving R about the line x=4.


a. What is the radius of a cylindrical shell at a point x in [0, 4]?

Textbook Question

Compressing and stretching a spring Suppose a force of 15 N is required to stretch and hold a spring 0.25 m from its equilibrium position.

b. How much work is required to compress the spring 0.2 m from its equilibrium position?