Calculus 3 Riemann Sum

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Each of those contour lines is giving you a constant value for your function $f(x,y)$ along that line. Presumably, from the way the picture is drawn, they're hoping you'll select the labeled $P_{ij}$ for the interval $[i-1,i]\times[j-1,j]$ in your Riemann sum. For example, since you have the value of $P_{21}$ is 3, a representative value for your sum is square $[0,1]\times[1,2]$ would be $P_{12}=3$ as well.

It looks as though the contour lines are increasing by one unit value each as you move up and to the right, so my guess would be that since the contour line that $P_{13}$ sits on has value $f(P_{13})=5$, the value at $P_{23}$ should be $f(P_{23})=6$, the value of $f$ at $P_{32}$ is 7, and the value at $P_{33}$ is 10.

So you've got a grid of unit squares with values for $f(x,y)$ at sample values of 2,3,3,4,4,5,6,7,10, letting you compute a Riemann sum for the integral of $f(x,y)$ over the region $[0,3]\times[0,3]$. This problem (WeBWoRK?) wants you to use this sum to compute the integral of $g(x,y)=f(x,y)+4$, which should be straightforward with the Riemann sum of $f(x,y)$ there.

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Updated on August 01, 2022

Comments

  • Christopher
    Christopher over 1 year

    enter image description here

    I do not really know where to start for this problem. I have solved ones with rectangular shape where the points were aligned but never a problem like this. If anyone could help me start this problem or give me steps to solving an answer I would be very thankful.

  • Christopher
    Christopher over 9 years
    Okay, I am starting to get it, but I am still confused as to what f(x,y) is.