Ch. 7 - Transcendental Functions

Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook

Hass 15th Edition

Ch. 7 - Transcendental Functions

Problem 7.AAE.1

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 guidance

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.

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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.

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.

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.

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Derivatives of Other Inverse Trigonometric Functions

Related Practice

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.

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Textbook Question

In Exercises 9 and 10, use implicit differentiation to find dy/dx.

9. y^e^x = x^y + 1

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Textbook Question

7. Let A(t) be the area of the region in the first quadrant enclosed by the coordinate axes, the curve y=e^(-x), and the vertical line x=t, t>0. Let V(t) be the volume of the solid generated by revolving the region about the x-axis. Find the following limits.

a. lim(x→∞)A(t)

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Textbook Question

In Exercises 25–30, use logarithmic differentiation to find the derivative of y with respect to the appropriate variable.

25. y = 2(x² + 1)/√(cos 2x)

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Textbook Question

7. What integrals lead to logarithms? Give examples. What are the integrals of tan x, cot x, sec x, and csc x?

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Textbook Question

111. True, or false? Give reasons for your answers.

e. arctan x = O(1)