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Equal integrals Without evaluating integrals, explain the following equalities. (Hint: Draw pictures.)
b. ∫²₀(25−(x²+1)²) dx = 2∫₁⁵ y√y−1 dy
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Ch. 6 - Applications of Integration
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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TOC.tableOfContents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Capitolo 6, Problema 6.1.59b
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
b. When the velocity is positive on an interval, the displacement and the distance traveled on that interval are equal.
Guida verificata passo dopo passo1
Recall the definitions: displacement on an interval is the net change in position, calculated as \(s(b) - s(a)\) where \(s(t)\) is the position function, and distance traveled is the total length of the path traveled, which accounts for all movement regardless of direction.
Understand that velocity being positive on an interval means the object is moving forward (increasing position) throughout that interval, so \(v(t) > 0\) for all \(t\) in the interval.
Since velocity is positive, the position function \(s(t)\) is strictly increasing on that interval, meaning the object never moves backward.
Because the object only moves forward, the displacement (net change in position) equals the total distance traveled, as there are no backward movements to add extra distance.
Therefore, when velocity is positive on an interval, displacement and distance traveled on that interval are indeed equal.
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Velocity and Its Significance
Velocity is the rate of change of position with respect to time and includes direction, indicated by its sign. A positive velocity means the object is moving in the positive direction, which affects how displacement and distance are calculated over an interval.
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Percorso guidato
06:29
Derivatives Applied To Velocity
Displacement vs. Distance Traveled
Displacement is the net change in position from the start to the end of an interval, considering direction, while distance traveled is the total length of the path covered, always non-negative. They are equal only if the motion does not change direction.
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Percorso guidato
08:56Using The Acceleration Function Example 1
Relationship Between Velocity and Displacement/Distance
If velocity remains positive over an interval, the object moves continuously in one direction, so displacement equals distance traveled. However, if velocity changes sign, displacement and distance differ, highlighting the importance of velocity's sign in comparing these quantities.
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Percorso guidato
06:29Derivatives Applied To Velocity
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Domanda del libro di testo
Calculating work for different springs Calculate the work required to stretch the following springs 0.4 m from their equilibrium positions. Assume Hooke’s law is obeyed.
b. A spring that requires 2 J of work to be stretched 0.1 m from its equilibrium position
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Domanda del libro di testo
Deceleration A car slows down with an acceleration of a(t) = −15 ft/s². Assume v(0)=60 ft/s,s(0)=0, and t is measured in seconds.
b. How far does the car travel in the time it takes to come to rest?
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13–16. Displacement from velocity Consider an object moving along a line with the given velocity v. Assume time t is measured in seconds and velocities have units of m/s.
b. Find the displacement over the given interval.
v(t) = 3t²−6t on [0, 3]
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Domanda del libro di testo
40–43. Population growth
When records were first kept (t=0), the population of a rural town was 250 people. During the following years, the population grew at a rate of P′(t) = 30(1+√t), where t is measured in years.
b. Find the population P(t) at any time t≥0.
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Domanda del libro di testo
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.
b. The distance traveled between t=0 and t=5
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