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Ch. 5 - 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. 5 - IntegrationProblema 5.1.71b
Capitolo 5, Problema 5.1.71b

Displacement from a velocity graph Consider the velocity function for an object moving along a line (see figure).
(b) Use geometry to find the displacement of the object between t = 0 and t = 2.
Graph showing velocity in meters per second over time in seconds, with a flat line indicating constant velocity segments.

Guida verificata passo dopo passo
1
Step 1: Understand the problem. The displacement of an object can be found by calculating the area under the velocity-time graph between t = 0 and t = 2 seconds. This is because displacement is the integral of velocity over time.
Step 2: Analyze the graph. Between t = 0 and t = 2 seconds, the velocity graph forms two geometric shapes: a triangle from t = 0 to t = 1 and a rectangle from t = 1 to t = 2.
Step 3: Calculate the area of the triangle. The triangle has a base of 1 second (from t = 0 to t = 1) and a height of 20 m/s (velocity at t = 1). Use the formula for the area of a triangle: A = (1/2) × base × height.
Step 4: Calculate the area of the rectangle. The rectangle spans from t = 1 to t = 2 seconds, with a width of 1 second and a constant height of 20 m/s. Use the formula for the area of a rectangle: A = width × height.
Step 5: Add the areas of the triangle and rectangle. The total displacement is the sum of these areas, which represents the total area under the velocity graph from t = 0 to t = 2 seconds.

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Velocity and Displacement

Velocity is the rate of change of displacement with respect to time, indicating how fast an object is moving in a specific direction. Displacement, on the other hand, is the overall change in position of the object, which can be calculated by integrating the velocity function over a given time interval. In this context, understanding the relationship between velocity and displacement is crucial for solving the problem.
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Percorso guidato
10:17
Using The Velocity Function

Area Under the Curve

In a velocity-time graph, the displacement of an object can be determined by calculating the area under the velocity curve between two time points. Each segment of the graph represents a different velocity, and the area can be computed using geometric shapes such as rectangles and triangles. This geometric approach simplifies the process of finding displacement without needing to perform calculus directly.
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Percorso guidato
05:59
Estimating the Area Under a Curve with Right Endpoints & Midpoint

Geometric Shapes in Graphs

When analyzing a velocity graph, different segments can form various geometric shapes, such as rectangles and triangles. The area of these shapes corresponds to the displacement during specific time intervals. For example, a rectangle's area is calculated as base times height, while a triangle's area is one-half base times height, allowing for straightforward calculations of displacement based on the graph's features.
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06:15
Graphing The Derivative
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