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Ch. 6 - Applications of Integration
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 6 - Applications of IntegrationProblema 6.2.5a
Capitolo 6, Problema 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).

Guida verificata passo dopo passo
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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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)
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Calculating work for different springs Calculate the work required to stretch the following springs 0.5m from their equilibrium positions. Assume Hooke’s law is obeyed.

a. A spring that requires a force of 50 N to be stretched 0.2 m from its equilibrium position

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a. What is the SAV ratio of a cube with side lengths a?

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9–10. Velocity graphs The figures show velocity functions for motion along a line. Assume the motion begins with an initial position of s(0)=0. Determine the following.

a. The displacement between t=0 and t=5

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Consider a solid whose base is the region in the first quadrant bounded by the curve y=√3−x and the line x=2, and whose cross sections through the solid perpendicular to the x-axis are squares.


a. Find an expression for the area A(x) of a cross section of the solid at a point x in [0, 2].

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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?

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Determine whether the following statements are true and give an explanation or counterexample.


a. The area of the region bounded by y=x and x=y^2 can be found only by integrating with respect to x.

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