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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 6.2.55
Capitolo 6, Problema 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

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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.

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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.
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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.
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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.
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Finding Area Between Curves on a Given Interval
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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 ________.


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


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

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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

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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. 


x = x³ ,y = 1, and x = 0; about the x-axis

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Look again at the region R in Figure 6.38 (p. 439). Explain why it would be difficult to use the washer method to find the volume of the solid of revolution that results when R is revolved about the y-axis.

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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.


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