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Chapter 7: Problem 15

(a) Use the End paper Integral Table to evaluate the given integral. (b) If you have a CAS, use it to evaluate the integral, and then confirm that the result is equivalent to the one that you found in part (a). $$\int \sqrt{9-x^{2}} d x$$

Short Answer

Expert verified
The integral evaluates to \( \frac{9}{2} \sin^{-1}\left(\frac{x}{3}\right) + \frac{x \sqrt{9-x^2}}{2} + C \).

Step by step solution

01

Recognize the Form of the Integral

Notice that the integral \( \int \sqrt{9-x^{2}} \, dx \) matches the form of a standard trigonometric substitution integral. In the End Paper Integral Table, look for integrals of the form \( \int \sqrt{a^2 - x^2} \, dx \).
02

Identify the Appropriate Substitution

For the integral \( \int \sqrt{9-x^{2}} \, dx \), note that \( a^2 = 9 \), hence \( a = 3 \). Use the substitution \( x = 3 \sin(\theta) \) so that \( dx = 3 \cos(\theta) \, d\theta \).
03

Simplify the Integral Using Trigonometric Identity

Substitute \( x = 3 \sin(\theta) \) into the integral. The expression \( \sqrt{9-x^2} \) becomes \( \sqrt{9 - 9 \sin^2(\theta)} = \sqrt{9 \cos^2(\theta)} = 3 \cos(\theta) \).
04

Change the Integral to Terms of Theta

Substitute \( x = 3 \sin(\theta) \) and \( dx = 3 \cos(\theta) \, d\theta \) into the integral: \[ \int \sqrt{9-x^2} \, dx = \int 3 \cos(\theta) \cdot 3 \cos(\theta) \, d\theta = 9 \int \cos^2(\theta) \, d\theta \].
05

Evaluate the Integral

Use the identity \( \cos^2(\theta) = \frac{1 + \cos(2\theta)}{2} \) to evaluate: \[ 9 \int \cos^2(\theta) \, d\theta = 9 \int \frac{1 + \cos(2\theta)}{2} \, d\theta = \frac{9}{2} \int (1 + \cos(2\theta)) \, d\theta \].
06

Solve the Integral

After integrating, you have: \( \frac{9}{2} \left( \theta + \frac{1}{2} \sin(2\theta) \right) + C \).
07

Back Substitute for x

Recall that \( x = 3 \sin(\theta) \), so \( \theta = \sin^{-1}\left(\frac{x}{3}\right) \). The solution becomes \( \frac{9}{2} \left( \sin^{-1}\left(\frac{x}{3}\right) + \sin(\theta) \cos(\theta) \right) + C \). Return to \( x \) by using \( \sin(\theta) = \frac{x}{3} \) and \( \cos(\theta) = \sqrt{1-\left(\frac{x}{3}\right)^2} = \frac{\sqrt{9-x^2}}{3} \).
08

Final Expression for the Integral

The final expression for the indefinite integral is: \[ \frac{9}{2} \sin^{-1}\left(\frac{x}{3}\right) + \frac{x \sqrt{9-x^2}}{2} + C \].
09

Using CAS to Evaluate the Integral

Use a Computer Algebra System (CAS) to solve \( \int \sqrt{9-x^2} \, dx \). This should confirm our manual solution: \( \frac{9}{2} \sin^{-1}\left(\frac{x}{3}\right) + \frac{x \sqrt{9-x^2}}{2} + C \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Integral Calculus
Integral calculus is a core aspect of calculus that deals with finding the total accumulation of quantities. At its essence, it's about summing infinite small amounts to find whole quantities, which is particularly useful in areas like physics for computing areas, volumes, and other accumulations. Integral calculus involves two main operations, definite integrals and indefinite integrals.
  • Definite Integrals: These are used to evaluate the accumulation of quantities over a specified interval. They give a numerical result and are used in calculating areas under curves. For example, you might use it to find the total distance traveled over a period when velocity is known.
  • Indefinite Integrals: These represent a family of functions and include a constant of integration, often denoted as +C. This is because the process of integration is the inverse of differentiation, so the original function can vary by a constant without affecting its derivative.
Trigonometric substitution, as used in the given problem, is a technique in integral calculus where trigonometric identities simplify the integration of certain algebraic functions.
Trigonometric Identities
Trigonometric identities are mathematical equations that express one trigonometric function in terms of others. They are quite essential in calculus, particularly in transformations and simplifications needed to solve integrals or derivatives. In the given problem, trigonometric identities play a vital role.When dealing with integrals involving square roots such as \( \sqrt{9-x^2} \), a trigonometric substitution like \( x = 3 \sin(\theta) \) can transform the expression into a simpler trigonometric form, utilizing identities:
  • Pythagorean Identities: Such as \( \sin^2(\theta) + \cos^2(\theta) = 1 \) are fundamental. In this example, recognizing that \( \sqrt{9 - 9\sin^2(\theta)} = 3\cos(\theta) \) utilizes \( \cos^2(\theta) = 1 - \sin^2(\theta) \).
  • Double Angle Formulas: Used to further simplify the integral, such as \( \cos^2(\theta) = \frac{1 + \cos(2\theta)}{2} \), which makes the integration process more straightforward.
Understanding and employing these identities allows conversion of complex integrals into more manageable forms.
Computer Algebra System (CAS)
A Computer Algebra System (CAS) is a software designed to perform symbolic mathematical computations. It can solve integrals, derivatives, and algebraic equations symbolically and is an invaluable tool in educational and professional settings.In the context of the given problem, a CAS can be used to solve the integral \( \int \sqrt{9-x^2} \, dx \) efficiently:
  • Verification: Confirming hand-computed integral solutions, like ensuring the result matches \( \frac{9}{2} \sin^{-1}\left(\frac{x}{3}\right) + \frac{x \sqrt{9-x^2}}{2} + C \).
  • Simplification: CAS tools can quickly reduce complex expressions using built-in algorithms and trigonometric identities, which can save time during exams or when tackling complex problems.
  • Learning Aid: By providing step-by-step solutions, a CAS can help students understand the process and identify where they might have gotten stuck.
Incorporating CAS not only complements manual calculation but enhances understanding and confidence in solving complex integral calculus problems.

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Most popular questions from this chapter

(a) Give a reasonable informal argument, based on areas, that explains why the integrals $$\int_{0}^{+\infty} \sin x d x \quad \text { and } \quad \int_{0}^{+\infty} \cos x d x$$ diverge. (b) Show that \(\int_{0}^{+\infty} \frac{\cos \sqrt{x}}{\sqrt{x}} d x\) diverges.

Suppose that during a period \(t_{0} \leq t \leq t_{1}\) years, a company has a continuous income stream at a rate of \(I(t)\) dollars per year at time \(t\) and that this income is invested at an annual rate of \(r \%,\) compounded continuously. The value (in dollars) of this income stream at the end of the time period \(t_{0} \leq t \leq t_{1},\) called the stream's future value, can be calculated using $$F V=\int_{t_{0}}^{t_{1}} I(t) e^{r\left(t_{1}-t\right)} d t$$ The present value (in dollars) of the income stream is given by $$P V=\int_{t_{0}}^{t_{1}} I(t) e^{-r\left(t-t_{0}\right)} d t$$ The present value is the amount that, if put in the bank at time \(t=t_{0}\) at \(r \%\) compounded continuously, with no additional deposits, would result in a balance of \(F V\) dollars at time \(t=t_{1}\) That is, $$F V=P V \times e^{r\left(t_{1}-t_{0}\right)}$$ In each exercise, (a) find the future value \(F V\) for the given income stream \(I(t)\) and interest rate \(r\) and time period \(t_{0} \leq t \leq t_{1}\) (b) find the present value \(P V\) of the income stream over the time period; and (c) verify that \(F V\) and \(P V\) satisfy the relationship given above. $$I(t)=2000 t+400 e^{-t} ; r=8 \% ; 0 \leq t \leq 10$$

Use any method to find the arc length of the curve. $$y=2 x^{2}, 0 \leq x \leq 2$$

Make the \(u\) -substitution and evaluate the resulting definite integral. $$\int_{12}^{+\infty} \frac{d x}{\sqrt{x}(x+4)} ; u=\sqrt{x} \quad[\text { Note: } u \rightarrow+\infty \text { as } x \rightarrow+\infty .]$$

Suppose that the region between the \(x\) -axis and the curve \(y=e^{-x}\) for \(x \geq 0\) is revolved about the \(x\) -axis. (a) Find the volume of the solid that is generated. (b) Find the surface area of the solid.
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