Chapter 6: Q. 47 (page 499)

Each definite integral represents the arc length of a function f(x) on an interval [a, b]. Determine the function and interval.
$\int_{- 2}^{5} \sqrt{1 + 9 e^{6 x}} d x$

Short Answer

Expert verified

The function is $f (x) = e^{3 x}$ and $[a, b] = [- 2, 5]$ .

Step by step solution

Step 1. Given information.

Consider the given function is $\int_{- 2}^{5} \sqrt{1 + 9 e^{6 x}} d x$ .

Step 2. Use arc length formula.

The formula for a function to find the arc length from $x = a$ to $x = b$ is given byrole="math" localid="1649233700729" $\int_{a}^{b} \sqrt{1 + {(f^{'} (x))}^{2}} d x$ .

Step 3. Find derivative of given function.

Compare given definite integral function with the arc length formula.

$\begin{array}{rcl} \int_{a}^{b} \sqrt{1 + {(f^{'} (x))}^{2}} d x & = & \int_{- 2}^{5} \sqrt{1 + 9 e^{6 x}} d x \\ = & \int_{- 2}^{5} \sqrt{1 + {(3 e^{3 x})}^{2}} d x \end{array}$

Therefore the function $f^{'} (x)$ will be $f^{'} (x) = 3 e^{3 x}$ and $[a, b] = [- 2, 5]$ .

Step 4. Find function f(x).

The function whose differentiation is $3 e^{3 x}$ is role="math" localid="1649234040169" $f (x) = e^{3 x}$ .

Thus the required function is $f (x) = e^{3 x}$ and $[a, b] = [- 2, 5]$ .

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.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Each definite integral represents the arc length of a function f(x) on an interval [a, b]. Determine the function and interval.
$\int_{- 2}^{5} \sqrt{1 + 9 e^{6 x}} d x$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Use arc length formula.

Step 3. Find derivative of given function.

Step 4. Find function f(x).

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Calculus

Mechanics Maths

Theoretical and Mathematical Physics

Decision Maths

Logic and Functions

Study anywhere. Anytime. Across all devices.

Each definite integral represents the arc length of a function f(x) on an interval [a, b]. Determine the function and interval.∫-251+9e6xdx

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Use arc length formula.

Step 3. Find derivative of given function.

Step 4. Find function f(x).

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Calculus

Mechanics Maths

Theoretical and Mathematical Physics

Decision Maths

Logic and Functions

Study anywhere. Anytime. Across all devices.

Each definite integral represents the arc length of a function f(x) on an interval [a, b]. Determine the function and interval.
$\int_{- 2}^{5} \sqrt{1 + 9 e^{6 x}} d x$