Textbook Question
9–40. Integration by parts Evaluate the following integrals using integration by parts.
26. ∫ t³ sin(t) dt
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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 8.7.42
41–48. Geometry problems Use a table of integrals to solve the following problems.
42. Find the length of the curve y = x^(3/2) + 8 on the interval from 0 to 2.
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Step 1: Recall the formula for the length of a curve. The length of a curve y = f(x) from x = a to x = b is given by: . This formula accounts for both the vertical and horizontal changes along the curve.
Step 2: Compute the derivative of y = x^(3/2) + 8 with respect to x. The derivative is: . The constant 8 disappears because its derivative is 0.
Step 3: Substitute the derivative into the curve length formula. The integrand becomes: . Simplify the square of the derivative term.
Step 4: Simplify the integrand further. The square of the derivative term becomes: . Thus, the integrand is: .
Step 5: Use a table of integrals to evaluate the integral of from x = 0 to x = 2. Look for a matching formula in the table of integrals and apply it to compute the curve length.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Arc Length Formula
The arc length formula is used to calculate the length of a curve defined by a function. For a function y = f(x), the length L from x = a to x = b is given by the integral L = ∫[a to b] √(1 + (dy/dx)²) dx. This formula incorporates the derivative of the function to account for the slope of the curve, providing an accurate measure of its length.
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Arc Length of Parametric Curves
Differentiation
Differentiation is the process of finding the derivative of a function, which represents the rate of change of the function with respect to its variable. In the context of the arc length formula, the derivative dy/dx is crucial as it helps determine how steep the curve is at any point, affecting the overall length calculation.
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Definite Integrals
Definite integrals are used to calculate the accumulation of quantities, such as area under a curve, over a specific interval [a, b]. In the context of finding arc length, the definite integral computes the total length of the curve between two points by summing infinitesimally small segments of the curve, providing a precise measurement of its extent.
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Related Practice
Textbook Question
Evaluate the following integrals.
∫ e³ˣ/(eˣ - 1) dx
Textbook Question
9–61. Trigonometric integrals Evaluate the following integrals.
16. ∫ sin²θ cos⁵θ dθ
Textbook Question
7–58. Improper integrals Evaluate the following integrals or state that they diverge.
44. ∫ (from 0 to ln 3) eʸ/(eʸ-1)⁷ᐟ³ dy
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Textbook Question
65-76. Volumes Find the volume of the described solid of revolution or state that it does not exist.
72. The region bounded by f(x) = (x + 1)^(-3/2) and the x-axis on the interval (-1, 1] is revolved about the line y = -1.
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Textbook Question
7–84. Evaluate the following integrals.
47. ∫ [(2x³ + x² - 2x - 4) / (x² - x - 2)] dx
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