Question

Consider the solid that lies above the square (in the xy-plane) R = [0,2] X [0,2], and below the elliptic paraboloid z=81−x^2−3y^2

Estimate the volume by dividing R into 4 equal squares and choosing the sample points to lie in the lower left hand corners.

Answer 1

try this
\[\frac{1}{16}\sum\limits_{i=0}^3\sum\limits_{j=0}^3 (81-(2i/4)^2-3(2j/4)^2) \]

Answer 2

http://www.wolframalpha.com/input/?i=%5Cfrac%7B1%7D%7B16%7D%5Csum%5Climits_%7Bi%3D0%7D%5E3%5Csum%5Climits_%7Bj%3D0%7D%5E3+%2881-%282i%2F4%29%5E2-3%282j%2F4%29%5E2%29+

Answer 3

@ganeshie8 77.5 is incorrect, i found a few yahoo answers examples, im looking at, I dont know what the question is asking, I don't know what R = [0,2] X [0,2] means

Answer 4

https://answers.yahoo.com/question/index?qid=20100328161647AAweAGd

Answer 5

that worked but i dont understand it still :(

Answer 6

Ahh okay, I have divided into four squares by mistake. try this :

\[\frac{1}{16}\sum\limits_{i=0}^1\sum\limits_{j=0}^1 (81-(2i/2)^2-3(2j/2)^2)\]

Answer 7

http://www.wolframalpha.com/input/?i=%5Cfrac%7B1%7D%7B16%7D%5Csum%5Climits_%7Bi%3D0%7D%5E1%5Csum%5Climits_%7Bj%3D0%7D%5E1+%2881-%282i%2F2%29%5E2-3%282j%2F2%29%5E2%29

Answer 8

that didnt work either, the correct answer is 216

Answer 9

what does R = [0,2] X [0,2] mean? I've never seen that notation before

Answer 10

it should be 316 right ?

Answer 11

http://www.wolframalpha.com/input/?i=%5Csum%5Climits_%7Bi%3D0%7D%5E1%5Csum%5Climits_%7Bj%3D0%7D%5E1+%2881-%282i%2F2%29%5E2-3%282j%2F2%29%5E2%29

Answer 12

oh yes 316 sorry

Answer 13

finally ! okay, R = [0,2] X [0,2]

make a guess what it could mean

Answer 14

a 2 by 2 square would be my guess

Answer 15

R = [0,2] X [0,2] gives the exact same info as bounds in a double integral : \(\large \int\limits_0^2\int\limits_0^2\)

```
x : 0 --> 2
y : 0--> 2
````

Answer 16

hmm ok