Question

calc help

Answer 1

are you able to graph the equations to see what this region looks like?

Answer 2

yes, i graphed it

Answer 3

the region we want to find is the red region

Answer 4

is there an equation to do that?

Answer 5

first we need to solve the given equation for x

get it in terms of y. We're doing this because it says to integrate with respect to y

Answer 6

x=4siny?

Answer 7

x = 4*sin(y)
yes

Answer 8

now think of x and y swapping places. That's effectively what happens when you do the inverse

Answer 9

if you graph x = 4*sin(y) like you would y = 4*sin(x), you get this

see attached

Answer 10

the region we want to find is this purple region

Answer 11

it turned on its side?

Answer 12

yes, basically the x axis turned into the y axis and vice versa

Answer 13

or as you put it, it reflected

Answer 14

so now what?

Answer 15

where does the region start and stop? in terms of y

Answer 16

0 and 4

Answer 17

imagine the y axis is now laying horizontally

Answer 18

it would look the same as the original graph?

Answer 19

0 and 1.5(ish)?

Answer 20

what do you get when you solve 4 = 4*sin(x) for x ?

Answer 21

do you divide by 4 first?

Answer 22

yes, so sin(x) = 1 which means x = ??

Answer 23

1.57

Answer 24

which is exactly pi/2

sin(pi/2) = 1

Answer 25

so we're integrating from y = 0 to y = pi/2

Answer 26

let

g(y) = 4
h(y) = 4*sin(y)

you want to find the value of
\[\Large \int_{0}^{\pi/2}\left[g(y)-h(y)\right]dy\]

Answer 27

so take the antiderivative then plug in 0 and pi/2, integrating

Answer 28

this will subtract out the green region that we don't want (Leaving just the purple region we do want)

Answer 29

`so take the antiderivative then plug in 0 and pi/2, integrating`
correct

Answer 30

i got 2.283

Answer 31

let

g(y) = 4
h(y) = 4*sin(y)
A = area of region we want


\[\Large A = \int_{0}^{\pi/2}\left[g(y)-h(y)\right]dy\]
\[\Large A = \int_{0}^{\pi/2}\left[4-4\sin(y)\right]dy\]
\[\Large A = 4x + 4\cos(y)\Bigg]_{0}^{\pi/2}\]
\[\Large A = \left(4\left(\frac{\pi}{2}\right) + 4\cos\left(\frac{\pi}{2}\right)\right)-\left(4\left(0\right) + 4\cos\left(0\right)\right)\]
\[\Large A = \left(2\pi + 4(0)\right)-\left(4\left(0\right) + 4(1)\right)\]
\[\Large A = 2\pi-4 \ \ {\color{blue}{\text{See note1}}}\]
\[\Large A \approx 2.28318530717959 \ \ {\color{blue}{\text{See note2}}}\]


Note1: this is the exact answer in terms of pi
Note2: this is the approximate answer. Round however you need to

Answer 32

`i got 2.283` you got the correct approximate answer @helpppppppppp

Answer 33

yay, tysm

Answer 34

you're welcome