Problem 78 - Thomas Calculus

Question

Evaluate the integrals in Exercises \(71-94\) $$ \int_{0}^{3 \sqrt{2} / 4} \frac{d s}{\sqrt{9-4 s^{2}}} $$

Step-by-Step Solution

Answer

\(\frac{\pi}{8}\)

Step 1: Identify the Integral Type

The integral \( \int \frac{ds}{\sqrt{9-4s^2}} \) is a trigonometric substitution integral. It resembles the integral formula \( \int \frac{dx}{\sqrt{a^2-x^2}} = \arcsin\left(\frac{x}{a}\right) + C \).

Step 2: Rewrite the Integral

Rewrite the integral in the form of the standard formula. Start by factoring the constant inside the square root: \( \int \frac{ds}{\sqrt{9 - 4s^2}} \) can be rewritten as \( \int \frac{ds}{\sqrt{3^2-(2s)^2}} \).

Step 3: Perform Trigonometric Substitution

Set \( 2s = 3\sin\theta \). Then \( ds = \frac{3}{2}\cos\theta\,d\theta \). The limits transform as well: when \( s=0, \theta = 0 \); when \( s = \frac{3\sqrt{2}}{4}, \theta = \frac{\pi}{4} \).

Step 4: Substitute and Simplify

Substitute into the integral: \( \int_{0}^{\frac{\pi}{4}} \frac{\frac{3}{2}\cos\theta\,d\theta}{\sqrt{9-9\sin^2\theta}} = \int_{0}^{\frac{\pi}{4}} \frac{\frac{3}{2}\cos\theta\,d\theta}{3\cos\theta} \). Simplify to get \( \int_{0}^{\frac{\pi}{4}} \frac{1}{2}\,d\theta \).

Step 5: Integrate

Now, integrate \( \int_{0}^{\frac{\pi}{4}} \frac{1}{2}\,d\theta \), which results in \( \frac{1}{2}[\theta]_{0}^{\frac{\pi}{4}} \).

Step 6: Evaluate the Integral

Evaluate at the limits: \( \frac{1}{2}\left(\frac{\pi}{4} - 0\right) = \frac{\pi}{8} \).

Key Concepts

Definite Integral

A definite integral is a mathematical concept used to calculate the signed area under a curve in a given interval on a graph. It is denoted by the integral sign with limits, representing an accumulation of quantities. In the context of the given problem, we are finding the definite integral of a particular function from 0 to \(3\sqrt{2}/4\). This means we are interested in the exact area between the curve \(\frac{1}{\sqrt{9-4s^2}}\) and the \(s\)-axis, restricted by these boundary values.

The process involves using limits to specify the beginning and end of the interval over which you want to calculate the area. These boundaries make the integration "definite." Unlike an indefinite integral, which results in a general antiderivative function plus a constant \(C\), a definite integral results in a specific numerical value. This is accomplished by evaluating the antiderivative at both the upper and lower limits and computing the difference.

Integration Techniques

Integration techniques are strategies used to simplify and solve integrals, especially when they are not straightforward. In this exercise, we employed several techniques, including substitution and using a trigonometric identity, to tackle the integral.

• Trigonometric substitution: This involves transforming the integral into a more convenient form using trigonometric identities. For square roots like in \(\sqrt{a^2-x^2}\), a common substitution might involve \(x = a\sin\theta\).
• Simplification: By rewriting the original integral using our substitution, we transformed it into a simple integral \(\int \frac{1}{2}\,d\theta\), which is easy to integrate.

Trigonometric substitution is particularly useful for dealing with integrals that involve the square roots of quadratic forms. By transforming the variables, you can simplify the form of the integral and, in some cases, directly reduce it to a basic trigonometric function.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all angles. In calculus, they are highly useful when faced with integrals involving square roots of terms like \(\sqrt{a^2 - x^2}\), as they help simplify complex expressions.

In the step-by-step solution, we used the identity \(\sin^2\theta + \cos^2\theta = 1\) to facilitate the trigonometric substitution process.

• By setting \(2s = 3\sin\theta\), we created a situation where \(\sqrt{9 - 9\sin^2\theta}\) could be simplified to \(3\cos\theta\), taking advantage of the Pythagorean identity.
• The trigonometric identity simplifies the expression and helps to directly solve the modified integral after the substitution \(\int \frac{3}{2}\cos\theta \,d\theta\) which then simplifies to \(\int \frac{1}{2} \,d\theta\).

These identities are pivotal in transforming and simplifying the original integral to a form that is easily integrable, which allows us to find the solution more efficiently.