Problem 4. Convert the integral -X to polar coordinates and evaluate it (use t for 0): and d = b= With a = √6 v6 fx, dydx = f fd C = _drdt = = Sa dy dx -" ib dt
Problem 4. Convert the integral -X to polar coordinates and evaluate it (use t for 0): and d = b= With a = √6 v6 fx, dydx = f fd C = _drdt = = Sa dy dx -" ib dt
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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1.4
please solve it on paper
![## Problem 4
Convert the integral
\[
\int_0^{\sqrt{6}} \int_{-x}^x dy dx
\]
to polar coordinates and evaluate it (use \( t \) for \( \theta \)):
With \( a = \) _____, \( b = \) _____, \( c = \) _____, and \( d = \) _____,
\[
\int_0^{\sqrt{6}} \int_{-x}^x dy dx = \int_a^b \int_c^d dr dt
\]
\[
\hspace{1em} = \int_a^b \hspace{1em} dt
\]
\[
\hspace{1em} = \left[ \hspace{10em} \right]_a^b
\]
\[
\hspace{1em} = \hspace{10em}.
\]
### Explanation
The problem involves converting a given double integral from Cartesian coordinates to polar coordinates and then evaluating it.
In Cartesian coordinates, the limits of integration are from \( y = -x \) to \( y = x \) and from \( x = 0 \) to \( x = \sqrt{6} \). The task is to find the corresponding limits in polar coordinates (\( r \) and \( \theta \)) and then rewrite and evaluate the integral.
The integral bounds for \( r \) and \( \theta \) need to be determined and will be substituted into the integral in the polar form.
The step-by-step process is indicated in sections where values \( a \), \( b \), \( c \), and \( d \), and the resulting substitutions, need to be filled in and calculated.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbbff2935-77bb-4550-bfd1-d595e6271f30%2Fd5d6e355-b8ab-4b5a-bed0-f00b472d9eae%2F42rdn5_processed.png&w=3840&q=75)
Transcribed Image Text:## Problem 4
Convert the integral
\[
\int_0^{\sqrt{6}} \int_{-x}^x dy dx
\]
to polar coordinates and evaluate it (use \( t \) for \( \theta \)):
With \( a = \) _____, \( b = \) _____, \( c = \) _____, and \( d = \) _____,
\[
\int_0^{\sqrt{6}} \int_{-x}^x dy dx = \int_a^b \int_c^d dr dt
\]
\[
\hspace{1em} = \int_a^b \hspace{1em} dt
\]
\[
\hspace{1em} = \left[ \hspace{10em} \right]_a^b
\]
\[
\hspace{1em} = \hspace{10em}.
\]
### Explanation
The problem involves converting a given double integral from Cartesian coordinates to polar coordinates and then evaluating it.
In Cartesian coordinates, the limits of integration are from \( y = -x \) to \( y = x \) and from \( x = 0 \) to \( x = \sqrt{6} \). The task is to find the corresponding limits in polar coordinates (\( r \) and \( \theta \)) and then rewrite and evaluate the integral.
The integral bounds for \( r \) and \( \theta \) need to be determined and will be substituted into the integral in the polar form.
The step-by-step process is indicated in sections where values \( a \), \( b \), \( c \), and \( d \), and the resulting substitutions, need to be filled in and calculated.
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