Question

Can i get a little help please? not sure what to do here.....

Calculate a Riemann sum S3,3 on the square R=[0,3]×[0,3] for the function g(x,y)=f(x,y)+2.
The contour plot of f(x,y) is shown in Figure 4.
Choose sample points and use the plot to find the values of f(x,y) at these points.
Use the values of f(x,y) to evaluate g(x,y) accordingly.

Answer 1

@ganeshie8, can you please help me again?

Answer 2

Notice that each square grid has an area of \(\Delta A = 1\)

Answer 3

You choose the sample points from \(P_{11}\) to \(\large P_{33}\) one in each grid

Answer 4

\[\large S_{3,3} = \sum \limits_{i=1}^3\sum\limits_{j=1}^3P_{ij}\cdot \Delta A\]

Answer 5

since the area in xy plane is a constant 1 :
\[\large S_{3,3} = \sum \limits_{i=1}^3\sum\limits_{j=1}^3P_{ij}\]

Answer 6

just find all the 9 values and add them up

Answer 7

trying to remember reimann sum. it's been a year since calc 1. i know you add up all the amounts and divide by the number of sections....

Answer 8

there is nothing to divide here, its more simpler than you think it is

Answer 9

we're just slicing a solid into pieces of square base of side 1,
finding each piece volume and adding up

Answer 10

http://math.oregonstate.edu/home/programs/undergrad/CalculusQuestStudyGuides/vcalc/255doub/plot4.gif

Answer 11

the volume of each slice is base area times height,
since the base is just an unit square, the volume simply equals the height (or value of the function g(x,y) )

Answer 12

so how do we determine the height with no points?

Answer 13

none of the point values are very clear

Answer 14

Look at the given contour plot,
\(\large P_{11} = f(x,y) + 2 = 2 + 2 = 4\)
\(\large P_{12} = f(x,y) + 2 = 3 + 2 = 5\)
\(\large P_{13} = f(x,y) + 2 = 5 + 2 = 7\)

Answer 15

the value of a function stays same on a level curve

Answer 16

oh. i'm blind. didn't see the 2 next to the points. lol.

Answer 17

:) from that you can assume that the level curves are incrasing their value as you move out

Answer 18

can you figure out remainign 6 values ?

Answer 19

wait, P (1,1) is 2 and P (2,1) is 3

Answer 20

wrong, P(1,1) is 2+2
and P(2,1) is 3+2

Answer 21

the value is for the whole curve

Answer 22

don't forget that the given contour plot is of f(x,y)
but you want the riemann sum for g(x,y)

Answer 23

so you need to add +2 to each value

Answer 24

i have P(1,1)=4. P (1,2)=5, P (2,1)=5, P(2,2)=5. P(3,1)=6, P(1,3)=7 and P (3,3) = 10. right?

Answer 25

er

Answer 26

wrong

Answer 27

check P(2,2) again

Answer 28

P(3,3) is 12.

Answer 29

P(2,3)=8, P(3,2)=9, P(3,1)=6. etc. P(2,2) = 6...oops

Answer 30

looks good, add them all up and you're done

Answer 31

so just add up all those numbers? 4+5+5+6+6+7+8+9+12?

Answer 32

62. correct answer! yay!!

Answer 33

good job!

Answer 34

kinda wish webwork gave hints.....are you a teacher, ganeshie? you're so very helpful.

Answer 35

actually explain things rather than just giving me an answer. i appreciate that.

Answer 36

np, yw :) calc3 is fun indeed! you will never be needing these riemann sums again.. calc3 is all about line integrals(work) and surface integrals(flux)

Answer 37

thanks, this is my second attempt to do calc 3.