Textbook Question
Area: Find the area enclosed by the ellipse x²/a² + y²/b² = 1.
1
views
Ch. 8 - Techniques of Integration
Hass15th EditionThomas' CalculusISBN: 9780137616077
Not the one you use?Change textbookSelect a different textbook
Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 8, Problem 8.8.16
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₀² (s + 1) / √(4 − s²) ds
Verified step by step guidance1
Identify the integral to be solved: \(\int_0^2 \frac{s + 1}{\sqrt{4 - s^2}} \, ds\).
Split the integral into two separate integrals for easier handling: \(\int_0^2 \frac{s}{\sqrt{4 - s^2}} \, ds + \int_0^2 \frac{1}{\sqrt{4 - s^2}} \, ds\).
For the first integral \(\int_0^2 \frac{s}{\sqrt{4 - s^2}} \, ds\), use the substitution method. Let \(u = 4 - s^2\), then find \(du\) and express \(s \, ds\) in terms of \(du\).
For the second integral \(\int_0^2 \frac{1}{\sqrt{4 - s^2}} \, ds\), recognize it as a standard integral form related to the arcsine function: \(\int \frac{1}{\sqrt{a^2 - x^2}} \, dx = \arcsin\left(\frac{x}{a}\right) + C\).
Evaluate both integrals using the substitution and standard integral results, then combine the results and apply the limits from 0 to 2 to find the value of the original integral.
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
11mWas this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Definite Integrals
A definite integral calculates the net area under a curve between two limits. It is evaluated by finding the antiderivative of the integrand and then applying the Fundamental Theorem of Calculus to compute the difference at the upper and lower bounds.
Recommended video:
05:43
Definition of the Definite Integral
Substitution Method
The substitution method simplifies integrals by changing variables to transform the integral into a more manageable form. It often involves setting a part of the integrand equal to a new variable, which helps in integrating complex expressions like those involving square roots.
Recommended video:
07:33Euler's Method
Integrals Involving Square Roots of Quadratic Expressions
Integrals with terms like √(a² − x²) often require trigonometric substitution or recognizing standard integral forms. These techniques help convert the integral into a trigonometric integral, which is easier to evaluate without tables.
Recommended video:
03:33Integrals Involving Natural Logs: Substitution Example 7
Related Practice
Textbook Question
For Exercises 49–52, complete the square before using an appropriate trigonometric substitution.
∫ 1 / √(x² - 2x + 5) dx
Textbook Question
The integrals in Exercises 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.
∫ 3 sinh(x/2 + ln 5) dx
1
views
Textbook Question
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₋₈¹ dx / x^(1/3)
1
views
Textbook Question
Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.
∫ (cos(√x))/(√x) dx
Textbook Question
Evaluate the integrals in Exercises 1–22.
∫ sin⁴(2x) cos(2x) dx