Textbook Question
17. Even-odd decompositions
b. If f(x) = f_E(x) + f_O(x) is the sum of an even function f_E(x) and an odd function f_O(x), then show that
f_E(x) = (f(x)+f(-x))/2 and f_O(x) = (f(x)-f(-x))/2
Ch. 7 - Transcendental Functions
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 7, Problem 7.AAE.1
Find the limits in Exercises 1–6.
1. lim(b→1⁻) ∫(from 0 to b) dx/√(1-x²)
Verified step by step guidance1
Recognize that the integral \( \int_0^b \frac{dx}{\sqrt{1 - x^2}} \) represents the inverse sine function, since \( \frac{d}{dx} \arcsin x = \frac{1}{\sqrt{1 - x^2}} \).
Rewrite the integral using the antiderivative: \( \int_0^b \frac{dx}{\sqrt{1 - x^2}} = \arcsin b - \arcsin 0 \).
Evaluate \( \arcsin 0 \), which is 0, so the integral simplifies to \( \arcsin b \).
Set up the limit expression as \( \lim_{b \to 1^-} \arcsin b \), where \( b \) approaches 1 from the left side.
Recall the value of \( \arcsin 1 \) and use it to determine the limit of the integral as \( b \) approaches 1 from the left.
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
3mWas this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Definite Integral as a Limit of Riemann Sums
A definite integral represents the accumulation of quantities, such as area under a curve, between two limits. It is defined as the limit of Riemann sums as the partition gets finer. Understanding this helps interpret the integral ∫₀ᵇ 1/√(1-x²) dx as a function of the upper limit b.
Recommended video:
06:11
Introduction to Riemann Sums
Limit of a Function from the Left (One-Sided Limit)
A left-hand limit, denoted as lim(b→1⁻), considers values of b approaching 1 from values less than 1. This concept is crucial when the function or integral behaves differently near the boundary, ensuring the limit is evaluated correctly as b approaches 1 from below.
Recommended video:
05:50One-Sided Limits
Integral of 1/√(1 - x²) and its Connection to Inverse Trigonometric Functions
The integral of 1/√(1 - x²) with respect to x is arcsin(x) + C. Recognizing this allows direct evaluation of the definite integral by applying the Fundamental Theorem of Calculus, simplifying the limit problem to evaluating arcsin(b) as b approaches 1 from the left.
Recommended video:
06:35Derivatives of Other Inverse Trigonometric Functions
Related Practice
Textbook Question
13. For what x>0 does x^(x^x) = (x^x)^x? Give reasons for your answer.
Textbook Question
Find the limits in Exercises 1–6.
3. lim(x→0⁺) (cox(√x))^(1/x)
Textbook Question
20. Solid of revolution The region between the curve y=1/(2√x) and the x-axis from x=1/4 to x=4 is revolved about the x-axis to generate a solid.
a. Find the volume of the solid.
Textbook Question
Find the areas between the curves y=2(log_2(x))/x and y=2(log_4(x))/x and the x-axis from x=1 to x=e. What is the ratio of the larger area to the smaller?
Textbook Question
7. What integrals lead to logarithms? Give examples. What are the integrals of tan x, cot x, sec x, and csc x?