Textbook Question
71. Different Methods
Let I = ∫ (x²)/(x + 1) dx.
b. Evaluate I by first performing long division on the integrand.
Ch. 8 - Integration Techniques
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- 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: 9780136847243Not the one you use?Change textbook
Chapter 8, Problem 8.5.93b
93. Three start-ups Three cars, A, B, and C, start from rest and accelerate along a line according to the following velocity functions:
vₐ(t) = 88t/(t + 1), v_B(t) = 88t²/(t + 1)², and v_C(t) = 88t²/(t² + 1).
b. Which car travels farthest on the interval 0 ≤ t ≤ 5?
Verified step by step guidance1
To determine which car travels the farthest on the interval \(0 \leq t \leq 5\), we need to find the total distance each car covers during this time. Since the velocity functions \(v_A(t)\), \(v_B(t)\), and \(v_C(t)\) are given, the distance traveled by each car is the integral of its velocity over the interval from 0 to 5.
Set up the definite integrals for each car's distance traveled:
For car A:
\[D_A = \int_0^5 \frac{88t}{t + 1} \, dt\]
For car B:
\[D_B = \int_0^5 \frac{88t^2}{(t + 1)^2} \, dt\]
For car C:
\[D_C = \int_0^5 \frac{88t^2}{t^2 + 1} \, dt\]
Next, evaluate each integral separately. This may involve techniques such as substitution or partial fraction decomposition depending on the integrand. For example, for \(D_A\), consider rewriting the integrand or using substitution \(u = t + 1\) to simplify the integral.
After finding the antiderivatives, apply the Fundamental Theorem of Calculus by evaluating each antiderivative at the upper limit \(t=5\) and subtracting the value at the lower limit \(t=0\) to find the exact distance traveled by each car.
Finally, compare the three distances \(D_A\), \(D_B\), and \(D_C\). The car corresponding to the greatest value among these integrals travels the farthest on the interval \(0 \leq t \leq 5\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Velocity and Displacement Relationship
Velocity is the rate of change of displacement with respect to time. To find the distance traveled over a time interval, you integrate the velocity function over that interval. This integral gives the total displacement, which is essential for comparing how far each car travels.
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06:29
Derivatives Applied To Velocity
Definite Integration
Definite integration calculates the exact area under a curve between two points, representing accumulated quantities like displacement from velocity. Evaluating the integral of each velocity function from t = 0 to t = 5 will determine the total distance each car covers.
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05:43Definition of the Definite Integral
Analyzing Rational Functions
The given velocity functions are rational expressions involving polynomials in numerator and denominator. Understanding their behavior, such as growth rates and limits, helps anticipate which car might travel farther before performing integration.
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6:04Intro to Rational Functions
Related Practice
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66. Let f(x) = cos(x²).
b. Calculate f''(x).
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{Use of Tech} Powers of sine and cosine It can be shown that
∫ from 0 to π/2 of sinⁿx dx = ∫ from 0 to π/2 of cosⁿx dx =
{
[1·3·5···(n-1)]/[2·4·6···n] · π/2 if n ≥ 2 is even
[2·4·6···(n-1)]/[3·5···n] if n ≥ 3 is odd
}
b. Evaluate the integrals with n = 10 and confirm the result.
Textbook Question
Gamma function The gamma function is defined by Γ(p) = ∫ from 0 to ∞ of x^(p-1) e^(-x) dx, for p not equal to zero or a negative integer.
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Textbook Question
Area and volume Consider the function f(x) = (9 + x²)^(-1/2) and the region R on the interval [0, 4] (see figure).
b. Find the volume of the solid generated when R is revolved about the x-axis.
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