Resources

Resources

Bytelearn - cat image with glasses

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

[-/1 Points]
DETAILS
MY NOTES
SCALCET9 15.XP.3.019.
Evaluate the iterated integral by converting to polar coordinates.

int_(0)^(1)int_(y)^(sqrt(2-y^(2)))7(x+y)dxdy

[-/ $1$ Points] $\newline$ DETAILS $\newline$ MY NOTES $\newline$ SCALCET $9$ $15$ .XP. $3$ . $019$ . $\newline$ Evaluate the iterated integral by converting to polar coordinates. $\newline$ $\int_{0}^{1} \int_{y}^{\sqrt{2-y^{2}}} 7(x+y) d x d y$

View step-by-step help

View step-by-step help

Transformations of linear functions

Full solution

Q. [-/

1

Points]

\newline

DETAILS

\newline

MY NOTES

\newline

SCALCET

9

15

.XP.

3

.

019

.

\newline

Evaluate the iterated integral by converting to polar coordinates.

\newline

\int_{0}^{1} \int_{y}^{\sqrt{2-y^{2}}} 7(x+y) d x d y

Set Up Integral: First, we need to set up the integral in polar coordinates. The limits of integration for $r$ will go from $0$ to the circle defined by the equation $x^2 + y^2 = 2$ . The limits for $\theta$ will be from $0$ to $\frac{\pi}{2}$ because we're integrating over the first quadrant.
Convert to Polar: The equation $x^2 + y^2 = 2$ in polar coordinates is $r^2 = 2$ , so $r$ goes from $0$ to $\sqrt{2}$ . The integrand $7(x+y)$ in polar coordinates is $7(r\cos(\theta) + r\sin(\theta)) = 7r(\cos(\theta) + \sin(\theta))$ .
Write Integral: Now we can write the integral in polar coordinates: $\int_{0}^{\pi/2}\int_{0}^{\sqrt{2}}7r(\cos(\theta) + \sin(\theta)) \, dr \, d\theta$ .
Integrate with Respect to r: Let's integrate with respect to $r$ first: $\int_{0}^{\pi/2}(\int_{0}^{\sqrt{2}}7r(\cos(\theta) + \sin(\theta)) \,dr) \,d\theta$ .
Evaluate Integral for $r$ : The integral of $7r(\cos(\theta) + \sin(\theta))$ with respect to $r$ is $\frac{7}{2}r^2(\cos(\theta) + \sin(\theta))$ , evaluated from $0$ to $\sqrt{2}$ .
Integrate with Respect to theta: Plugging in the limits of integration for $r$ , we get: $\frac{7}{2}(2)(\cos(\theta) + \sin(\theta)) - 0 = 7(\cos(\theta) + \sin(\theta))$ .
Evaluate Integral for $\theta$ : Now we integrate $7(\cos(\theta) + \sin(\theta))$ with respect to $\theta$ from $0$ to $\pi/2$ : $\int_{0}^{\pi/2}7(\cos(\theta) + \sin(\theta)) \, d\theta.$
Simplify Final Result: The integral of $7\cos(\theta)$ is $7\sin(\theta)$ , and the integral of $7\sin(\theta)$ is $-7\cos(\theta)$ , both evaluated from $0$ to $\pi/2$ .
Simplify Final Result: The integral of $7\cos(\theta)$ is $7\sin(\theta)$ , and the integral of $7\sin(\theta)$ is $-7\cos(\theta)$ , both evaluated from $0$ to $\frac{\pi}{2}$ . Plugging in the limits of integration for $\theta$ , we get: $\newline$ $7\sin(\frac{\pi}{2}) - 7\sin(0) - (7\cos(\frac{\pi}{2}) - 7\cos(0))$ .
Simplify Final Result: The integral of $7\cos(\theta)$ is $7\sin(\theta)$ , and the integral of $7\sin(\theta)$ is $-7\cos(\theta)$ , both evaluated from $0$ to $\pi/2$ . Plugging in the limits of integration for $\theta$ , we get: $7\sin(\pi/2) - 7\sin(0) - (7\cos(\pi/2) - 7\cos(0))$ . Simplifying, we have: $7(1) - 7(0) - (7(0) - 7(1)) = 7 - 0 - (0 - 7) = 7 + 7 = 14$ .

More problems from Transformations of linear functions

Question

This equation shows how the time required to screen-print a batch of shirts is related to the number of shirts in the batch.

\newline

t = s + 13

\newline

s

represents the number of shirts in the batch, and the variable

t

represents the time required to screen-print the shirts. How long does it take to screen-print a batch of

4

\newline

_________minutes

Posted 3 months ago

Question

Krysta wants to take piano lessons. She asked a teacher how many new pieces she could expect to learn each month. The teacher gave an estimate of

3

. Write an equation that shows how the total number of pieces Krysta would learn,

y

, depends on the number of months of piano lessons,

x

\newline

y

Posted 3 months ago

Question

Tom bought a boat for

\$3,000

and incurs a daily storage fee of

\$15

. Write a linear function to represent the total amount Tom will owe,

p(x)

x

days to pick it up.

\newline

Posted 4 months ago

Question

180

-page book cover to cover in a single session, at a constant rate. It took him

6

\newline

\newline

y

represent the number of pages left to read after

\newline

t

\newline

Which of the following information about the graph of the relationship is given?

\newline

1

\newline

\newline

x

\newline

\newline

y

\newline

(C) Slope and a point that is not an intercept

\newline

\newline

x

\newline

y

\newline

\newline

y

-intercept and a point that is not an intercept

\newline

(F) Two points that are not intercepts

Posted 4 months ago

Question

A charity organization had to sell

18

tickets to their fundraiser just to cover necessary production costs. They sold each ticket for

\$45

y

represent the net profit (in dollars) when they have sold

x

tickets. Which of the following information about the graph of the relationship is given?

\newline

1

\newline

x

\newline

y

\newline

(C) Slope and a point that is not an intercept

\newline

x

y

\newline

y

-intercept and a point that is not an intercept

\newline

(F) Two points that are not intercepts

Posted 2 months ago

Question

Hiro painted his room. After

3

hours of painting at a rate of

8

square meters per hour, he had

28

square meters left to paint.

\newline

\newline

y

represent the area (in square meters) left to paint after

\newline

x

\newline

Which of the following information about the graph of the relationship is given?

\newline

1

\newline

\newline

x

\newline

\newline

y

\newline

(C) Slope and a point that is not an intercept

\newline

\newline

x

\newline

y

\newline

\newline

y

-intercept and a point that is not an intercept

\newline

(F) Two points that are not intercepts

Posted 2 months ago

Question

Naoya read a book cover to cover in a single session, at a rate of

55

pages per hour. After

4

350

pages left to read.

\newline

y

represent the number of pages left to read after

x

\newline

Complete the equation for the relationship between the number of pages left and number of hours.

\newline

y=

Posted 1 month ago

Question

A teacher purchased

300

colored pencils for an upcoming project. The pencils she ordered come in packs of

20

30

14

packs in all. Write a system of equations to find the number of

20

(x)

30

(y)

the art teacher ordered.

Posted 2 months ago

Question

C=0.25(L-1200)

\newline

C

, in dollars, to the holder of a liability insurance account given a total liability of

L

dollars is given by the equation as long as

C

is positive. By how many dollars does the liability increase when the cost to the holder increases by

1

Posted 2 months ago

Question

P=\frac{2}{5} m+1.5

\newline

P

, in dollars, of a train ticket used to travel a distance of

m

miles is given by the equation. By how many miles does the traveling distance increase if the ticket price increases by

1

Posted 4 months ago