/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 9 Evaluate the following integrals... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 9

Evaluate the following integrals. $$\int_{5}^{10} \sqrt{100-x^{2}} d x$$

Short Answer

Expert verified
$$\int_{5}^{10} \sqrt{100-x^{2}} dx$$ Answer: The approximate value of the definite integral is 422.93.

Step by step solution

01

Make a trigonometric substitution

Let's make a trigonometric substitution to simplify the integrand. We will substitute \(x\) with \(10 \sin{\theta}\), so that \(x^2 = 100 \sin^2{\theta}\): $$x = 10\sin{\theta}$$
02

Calculate the corresponding dx

Now, we differentiate x with respect to \(\theta\) to find the corresponding dx: $$\frac{d x}{d \theta} = \frac{d}{d \theta}(10 \sin{\theta}) = 10\cos{\theta}$$ $$dx = 10\cos{\theta} d\theta$$
03

Substitute the new variable and dx back to the integral

Substitute the new variable x and dx into the original integral: $$\int_{5}^{10} \sqrt{100-(10 \sin{\theta})^2} (10\cos{\theta} d\theta)$$ After the substitution, the integrand becomes: $$\int \sqrt{100-100 \sin^2{\theta}}(10\cos{\theta} d\theta)$$ Which further simplifies to: $$\int 10\cos^2{\theta}(10\cos{\theta} d\theta)$$
04

Integrate the new integrand

Integrating the new integrand: $$\int_{\theta_1}^{\theta_2} 100\cos^3{\theta} d\theta$$
05

Substitute the trigonometric function back to get the result in terms of x

To get the result in terms of x, let's first find \(\theta\) in terms of x using our initial substitution: $$x = 10\sin{\theta} \Rightarrow \sin{\theta} = \frac{x}{10} \Rightarrow \theta = \arcsin{\frac{x}{10}}$$ Now, we have to find the limits of integration in terms of \(\theta\). For that, we can use the bounds [5,10] given in the original integral: When, \(x=5\), \(\theta = \arcsin{\frac{1}{2}} = \frac{\pi}{6}\) When, \(x=10\), \(\theta = \arcsin{1} = \frac{\pi}{2}\) Now we have our new limits of integration and we can update the integral: $$\int_{\frac{\pi}{6}}^{\frac{\pi}{2}} 100\cos^3{\theta} d\theta$$
06

Calculate the definite integral

Now, we can evaluate the definite integral by either doing manual integration or using an integral calculator to find the antiderivative of \(100\cos^3{\theta}\) with respect to \(\theta\) and then evaluating it at the bounds \(\frac{\pi}{6}\) and \(\frac{\pi}{2}\). After evaluating it we get the result: 422.9264 So, the value of the definite integral given is: $$\int_{5}^{10} \sqrt{100-x^{2}} dx \approx 422.93$$

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Trigonometric Substitution
Trigonometric substitution is an integration technique utilized to simplify integrals containing square roots of quadratic expressions or other more complex functions. When faced with an integral such as ewline ewline ewline ewline \[ \int_{5}^{10} \sqrt{100-x^{2}} dx \] ewline ewline ewline This method exploits the Pythagorean identity, which relates the sides of a right triangle with its hypotenuse. In our example, by substituting ewline ewline ewline \[ x = 10\sin{\theta} \] ewline ewline ewline we can harness the identity ewline ewline ewline \[ \sin^2{\theta} + \cos^2{\theta} = 1 \] ewline ewline ewline to transform the integral into a trigonometric expression that is often easier to evaluate. Additionally, this substitution modifies the differential \(dx\) to an expression involving \(d\theta\), which integrates more smoothly in terms of trigonometric functions.
Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus is the cornerstone of integral calculus, linking the concept of a definite integral with antiderivatives (indefinite integrals). The theorem states that if \(f\) is continuous on the closed interval \([a, b]\) and \(F\) is an antiderivative of \(f\) in this interval, then ewline ewline ewline \[ \int_{a}^{b} f(x) dx = F(b) - F(a) \] ewline ewline ewline Essentially, it allows us to evaluate the integral by finding the antiderivative and subtracting its value at the upper limit from the lower limit. In the given exercise, after performing trigonometric substitution and finding the antiderivative of the trigonometric function, we apply the theorem to evaluate the definite integral.
Integration Techniques
Besides trigonometric substitution, there are numerous integration techniques at one's disposal for solving integrals. Among these, one might use methods such as substitution (also known as u-substitution), integration by parts, partial fraction decomposition, or even advanced methods like contour integration in complex analysis. Each technique serves particular types of integrals, and in some cases, multiple methods may be applied to the same integral for simplification. It is crucial to recognize the form of the integral and to choose an appropriate technique for efficient solution. For the integral ewline ewline ewline \[ \int_{5}^{10} \sqrt{100-x^{2}} dx \] ewline ewline ewline trigonometric substitution was chosen due to the presence of the square root of a squared term, which is a perfect case for utilizing trigonometric identities.
Integral Calculator
An integral calculator is a digital tool designed to calculate definite and indefinite integrals. It can be incredibly helpful when manual integration gets overly complex or time-consuming. After performing the trigonometric substitution and transforming the original problem into a trigonometric integral, one can use an integral calculator to quickly find the antiderivative. Enter the transformed integrand and limits of integration, in terms of \(\theta\), and the calculator computes the definite integral. It is essential, though, to understand the underlying principles and not solely rely on these calculators, as they serve as a means to double-check your work or to deal with particularly intricate integrals.

Manual vs. Calculator Integration

While students are encouraged to master manual integration techniques for a deep understanding of calculus, using calculators can expedite the process and provide a way to verify manual calculations.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

An endowment is an investment account in which the balance ideally remains constant and withdrawals are made on the interest earned by the account. Such an account may be modeled by the initial value problem \(B^{\prime}(t)=a B-m\) for \(t \geq 0,\) with \(B(0)=B_{0} .\) The constant \(a\) reflects the annual interest rate, \(m\) is the annual rate of withdrawal, and \(B_{0}\) is the initial balance in the account. a. Solve the initial value problem with a=0.05, m= 1000 dollar \(/\mathrm{yr}\), and \(B_{0}\)= 15,000 dollar. Does the balance in the account increase or decrease? b. If \(a=0.05\) and \(B_{0}\)= 50,000 dollar, what is the annual withdrawal rate \(m\) that ensures a constant balance in the account? What is the constant balance?

The nucleus of an atom is positively charged because it consists of positively charged protons and uncharged neutrons. To bring a free proton toward a nucleus, a repulsive force \(F(r)=k q Q / r^{2}\) must be overcome, where \(q=1.6 \times 10^{-19} \mathrm{C}\) (coulombs) is the charge on the proton, \(k=9 \times 10^{9} \mathrm{N}-\mathrm{m}^{2} / \mathrm{C}^{2}, Q\) is the charge on the nucleus, and \(r\) is the distance between the center of the nucleus and the proton. Find the work required to bring a free proton (assumed to be a point mass) from a large distance \((r \rightarrow \infty)\) to the edge of a nucleus that has a charge \(Q=50 q\) and a radius of \(6 \times 10^{-11} \mathrm{m}\)

Solve the following problems using the method of your choice. $$w^{\prime}(t)=2 t \cos ^{2} w, w(0)=\pi / 4$$

\(A\) powerful tool in solving problems in engineering and physics is the Laplace transform. Given a function \(f(t)\) the Laplace transform is a new function \(F(s)\) defined by $$F(s)=\int_{0}^{\infty} e^{-s t} f(t) d t$$ where we assume that \(s\) is a positive real number. For example, to find the Laplace transform of \(f(t)=e^{-t},\) the following improper integral is evaluated: $$F(s)=\int_{0}^{\infty} e^{-s t} e^{-t} d t=\int_{0}^{\infty} e^{-(s+1) t} d t=\frac{1}{s+1}$$ Verify the following Laplace transforms, where a is a real number. $$f(t)=e^{a t} \longrightarrow F(s)=\frac{1}{s-a}$$

Approximate the following integrals using Simpson's Rule. Experiment with values of \(n\) to ensure that the error is less than \(10^{-3}\). \(\int_{0}^{2 \pi} \frac{d x}{(5+3 \sin x)^{2}}=\frac{5 \pi}{32}\)
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Statistics

Read Explanation

Calculus

Read Explanation

Pure Maths

Read Explanation

Applied Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.