Hello...i just had a quick question...its probably something easy to get...maybe i am thiking too hard...but the question is estimate the volume of solid that likes above the square R=[0,2] * [0,2] and below the elliptical paraboloid z=16-x^2-2y^2..divide r into four equal squares and choose the sample point to be the upper right corner...i know how to find the volume
the formula for volume is v=sum of sequance f(x,y)A
i just need help on finding the area...then from then i can get the answer...on the book it says the area is 1...how is it one...is there a formula? plz help...the answer the probably right infront of me...maybe i am not seeing it...tnx for ur help in advance.
To find the volume of the solid that lies above the square R=[0,2] x [0,2] and below the elliptical paraboloid z=16-x^2-2y^2, you need to divide the square R into four equal squares and choose the sample point to be the upper right corner.
To estimate the volume using the formula v = sum of sequence f(x,y)A, you need to find the area A of each square and the corresponding function value f(x,y) at the sample point.
In this case, the book states that the area of each square is 1. This is because, when the square R=[0,2] x [0,2] is divided into four equal squares, each square has sides of length 1, resulting in an area of 1.
As for the function value f(x,y) at the sample point, it should be calculated as z=16-x^2-2y^2 at the upper right corner of each square. Substituting the coordinates of the upper right corner, which is (1,1) for each square, you will have f(1,1) = 16 - 1^2 - 2(1^2) = 16 - 1 - 2 = 13.
Therefore, for each square, the area A is 1 and the function value f(x,y) at the sample point is 13.
To estimate the volume, you need to sum up the product f(x,y)A for each square. Since there are four squares in total, the volume would be:
v = f(1,1)A + f(1,1)A + f(1,1)A + f(1,1)A
= 13(1) + 13(1) + 13(1) + 13(1)
= 13 + 13 + 13 + 13
= 52
So, the estimated volume of the solid is 52.
Hello! I can help you with finding the area of the square R=[0,2] * [0,2].
The area of a square is calculated by multiplying the length of one side by itself. In this case, the length of one side of the square is 2 units. Therefore, the area of the square is:
Area = 2 * 2 = 4 square units.
So, the area of square R=[0,2] * [0,2] is 4 square units.
I hope that helps! Let me know if you have any other questions.