Chapter 6: Q. 48 (page 499)

Each definite integral represents the arc length of a function f(x) on an interval [a, b]. Determine the function and interval.
$\int_{0}^{π} \sqrt{1 + \sin^{2} x} d x$

Short Answer

Expert verified

The function is $f (x) = - \cos x$ and $[a, b] = [0, π]$ .

Step by step solution

Step 1. Given information.

Consider the given function is $\int_{0}^{π} \sqrt{1 + \sin^{2} x} d x$ .

Step 2. Use arc length formula.

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

Step 3. Find derivative of given function.

Compare given definite integral function with the arc length formula.

$\begin{array}{rcl} \int_{a}^{b} \sqrt{1 + {(f^{'} (x))}^{2}} d x & = & \int_{0}^{π} \sqrt{1 + \sin^{2} x} d x \\ = & \int_{0}^{π} \sqrt{1 + {(\sin x)}^{2}} d x \end{array}$

Therefore the function $f^{'} (x) = \sin x$ androle="math" localid="1649235589478" $[a, b] = [0, π]$ .

Step 4. Find function fx.

The function whose differentiation is $\sin x$ is role="math" localid="1649235561134" $f (x) = - \cos x$ .

Thus the required function is $f (x) = - \cos x$ and $[a, b] = [0, π]$ .

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Each definite integral represents the arc length of a function f(x) on an interval [a, b]. Determine the function and interval.
$\int_{0}^{π} \sqrt{1 + \sin^{2} x} d x$

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 fx.

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Each definite integral represents the arc length of a function f(x) on an interval [a, b]. Determine the function and interval.∫0π1+sin2xdx

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 fx.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Mechanics Maths

Discrete Mathematics

Statistics

Probability and Statistics

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.
$\int_{0}^{π} \sqrt{1 + \sin^{2} x} d x$