Skip to main content
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 6.2.21
Capitolo 6, Problema 6.2.21

Determine the area of the shaded region in the following figures.
Graph showing a shaded region between two curves, with axes labeled x and y, indicating area calculation.

Guida verificata passo dopo passo
1
Step 1: Identify the curves that bound the shaded region. The graph shows the shaded region enclosed by the curves y = 2 - x/2 (a straight line) and x = 2y^2 (a parabola).
Step 2: Determine the points of intersection of the two curves. To find these points, solve the equations y = 2 - x/2 and x = 2y^2 simultaneously. Substitute x = 2y^2 into y = 2 - x/2 to find the y-values of the intersection points.
Step 3: Set up the integral to calculate the area. Since the region is bounded vertically (y-values), express x in terms of y for both curves. The parabola x = 2y^2 represents the left boundary, and the line x = 2 - y/2 represents the right boundary. The area is given by the integral of (right curve - left curve) with respect to y.
Step 4: Determine the limits of integration. The limits are the y-values of the intersection points found in Step 2. These limits define the range over which the area is calculated.
Step 5: Write the integral for the area. The area is given by the integral A = ∫[y1 to y2] [(2 - y/2) - (2y^2)] dy, where y1 and y2 are the limits of integration. Simplify the integrand and evaluate the integral to find the area.

Risposta video verificata per un problema simile:

Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.
Durata del video:
4m

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

Area Between Curves

The area between two curves can be calculated using definite integrals. To find this area, one must identify the upper and lower functions over the interval of interest. The area is given by the integral of the difference between the upper curve and the lower curve, integrated with respect to the variable of interest, typically x or y.
Video consigliato:
05:23
Finding Area Between Curves on a Given Interval

Definite Integrals

Definite integrals represent the accumulation of quantities, such as area under a curve, over a specified interval. The integral is evaluated using the Fundamental Theorem of Calculus, which connects differentiation and integration. The limits of integration correspond to the points where the curves intersect, defining the bounds of the area to be calculated.
Video consigliato:
Percorso guidato
05:43
Definition of the Definite Integral

Finding Points of Intersection

To determine the area between two curves, it is essential to find their points of intersection. This involves solving the equations of the curves simultaneously to identify the x-values where they meet. These intersection points serve as the limits of integration for calculating the area, ensuring that the correct region is being evaluated.
Video consigliato:
04:50
Critical Points
Pratica correlata
Domanda del libro di testo

Find the area of the surface generated when the given curve is revolved about the given axis.

x=4y^3/2−y^1/2 / 12, for 1≤y≤4; about the y-axis

60
views
Domanda del libro di testo

Find the arc length of the line y = 2x+1 on [1, 5] using calculus and verify your answer using geometry.

63
views
Domanda del libro di testo

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=|x| and y=2−x^2; about the x-axis 

52
views
Domanda del libro di testo

Determine the area of the shaded region in the following figures.

184
views
Domanda del libro di testo

Force on a triangular plate A plate shaped like an isosceles triangle with a height of 1 m is placed on a vertical wall 1 m below the surface of a pool filled with water (see figure). Compute the force on the plate.

87
views
Domanda del libro di testo

Determine the area of the shaded region in the following figures.

127
views