Values of f(x,y) are given in the table below. Let R be the rectangle 1≤x≤1.6, 2≤y≤3.2. Find a Riemann sum which is a reasonable estimate for ∫Rf(x,y)da with Δx=0.2 and Δy=0.4. Note that the values given in the table correspond to midpoints.
--------------x=1.1 --- x=1.3 --- x=1.5
y=2.2-----6 ----------6---------10
y=2.5-----8----------(-5)---------2
y=3-----(-5)-----------3---------10
Why did the rectangle go to the party? Because it wanted to estimate some integrals!
Let's calculate our Riemann sum step by step. With Δx = 0.2 and Δy = 0.4, we can divide our rectangle R into smaller sub-rectangles.
First, let's label the midpoints of each sub-rectangle on the x-axis: x = 1.1, 1.3, and 1.5.
Next, let's label the midpoints of each sub-rectangle on the y-axis: y = 2.2, 2.5, and 3.0.
Now, we can calculate the values of f(x,y) for each sub-rectangle. Let me write them down for you:
For the sub-rectangle with x = 1.1 and y = 2.2, f(1.1, 2.2) = 6.
For the sub-rectangle with x = 1.1 and y = 2.5, f(1.1, 2.5) = 8.
For the sub-rectangle with x = 1.1 and y = 3.0, f(1.1, 3.0) = -5.
For the sub-rectangle with x = 1.3 and y = 2.2, f(1.3, 2.2) = 6.
For the sub-rectangle with x = 1.3 and y = 2.5, f(1.3, 2.5) = -5.
For the sub-rectangle with x = 1.3 and y = 3.0, f(1.3, 3.0) = 3.
For the sub-rectangle with x = 1.5 and y = 2.2, f(1.5, 2.2) = 10.
For the sub-rectangle with x = 1.5 and y = 2.5, f(1.5, 2.5) = 2.
For the sub-rectangle with x = 1.5 and y = 3.0, f(1.5, 3.0) = 10.
Now, we can calculate the areas of each sub-rectangle by multiplying Δx and Δy:
For the sub-rectangle with x = 1.1 and y = 2.2, its area is 0.2 * 0.4 = 0.08.
For the sub-rectangle with x = 1.1 and y = 2.5, its area is 0.2 * 0.4 = 0.08.
For the sub-rectangle with x = 1.1 and y = 3.0, its area is 0.2 * 0.4 = 0.08.
For the sub-rectangle with x = 1.3 and y = 2.2, its area is 0.2 * 0.4 = 0.08.
For the sub-rectangle with x = 1.3 and y = 2.5, its area is 0.2 * 0.4 = 0.08.
For the sub-rectangle with x = 1.3 and y = 3.0, its area is 0.2 * 0.4 = 0.08.
For the sub-rectangle with x = 1.5 and y = 2.2, its area is 0.2 * 0.4 = 0.08.
For the sub-rectangle with x = 1.5 and y = 2.5, its area is 0.2 * 0.4 = 0.08.
For the sub-rectangle with x = 1.5 and y = 3.0, its area is 0.2 * 0.4 = 0.08.
Now, we can sum up the products of the values of f(x,y) and the areas of the sub-rectangles:
Riemann sum = (6 * 0.08) + (8 * 0.08) + (-5 * 0.08) + (6 * 0.08) + (-5 * 0.08) + (3 * 0.08) + (10 * 0.08) + (2 * 0.08) + (10 * 0.08)
I'll let you do the math on that one!
To find a Riemann sum for ∫Rf(x,y)da, we need to divide the rectangle R into smaller sub-rectangles and evaluate the function at a point within each sub-rectangle.
The width of each sub-rectangle, Δx, is given as 0.2, and the height of each sub-rectangle, Δy, is given as 0.4.
Let's start by dividing the x-interval 1 ≤ x ≤ 1.6 into sub-intervals:
1 ≤ x ≤ 1.2
1.2 ≤ x ≤ 1.4
1.4 ≤ x ≤ 1.6
Similarly, let's divide the y-interval 2 ≤ y ≤ 3.2 into sub-intervals:
2 ≤ y ≤ 2.4
2.4 ≤ y ≤ 2.8
2.8 ≤ y ≤ 3.2
Now, we can find the midpoints of each sub-rectangle:
For the x-values:
Midpoint of 1 ≤ x ≤ 1.2: (1 + 1.2) / 2 = 1.1
Midpoint of 1.2 ≤ x ≤ 1.4: (1.2 + 1.4) / 2 = 1.3
Midpoint of 1.4 ≤ x ≤ 1.6: (1.4 + 1.6) / 2 = 1.5
For the y-values:
Midpoint of 2 ≤ y ≤ 2.4: (2 + 2.4) / 2 = 2.2
Midpoint of 2.4 ≤ y ≤ 2.8: (2.4 + 2.8) / 2 = 2.6
Midpoint of 2.8 ≤ y ≤ 3.2: (2.8 + 3.2) / 2 = 3
Now, we can find the values of f(x, y) at each midpoint:
f(1.1, 2.2) = 6
f(1.1, 2.6) = 8
f(1.1, 3) = -5
f(1.3, 2.2) = 6
f(1.3, 2.6) = -5
f(1.3, 3) = 3
f(1.5, 2.2) = 10
f(1.5, 2.6) = 2
f(1.5, 3) = 10
Now, we can calculate the Riemann sum:
ΔA = Δx * Δy = 0.2 * 0.4 = 0.08
Riemann sum = Σ f(xi, yj) * ΔA
where Σ is the summation symbol, xi represents the x-values, yj represents the y-values, and ΔA is the area of each sub-rectangle.
Now, let's plug in the values:
Riemann sum = [f(1.1, 2.2) + f(1.1, 2.6) + f(1.1, 3)]*0.08
+ [f(1.3, 2.2) + f(1.3, 2.6) + f(1.3, 3)]*0.08
+ [f(1.5, 2.2) + f(1.5, 2.6) + f(1.5, 3)]*0.08
= [(6 + 8 + (-5))] * 0.08
+ [(6 + (-5) + 3)] * 0.08
+ [(10 + 2 + 10)] * 0.08
Simplifying further:
Riemann sum = [9] * 0.08 + [4] * 0.08 + [22] * 0.08
= 0.72 + 0.32 + 1.76
= 2.8
Therefore, a reasonable estimate for ∫Rf(x, y)da is 2.8.
To approximate the integral ∫Rf(x,y)da using a Riemann sum, we need to divide the region R into smaller rectangles and evaluate f(x, y) at a certain point in each rectangle.
First, let's find the number of subintervals in the x-direction and the y-direction. Given Δx = 0.2 and Δy = 0.4, we can calculate the number of subintervals as follows:
Number of subintervals in x-direction: (1.6 - 1) / 0.2 = 3
Number of subintervals in y-direction: (3.2 - 2) / 0.4 = 5
Now, let's divide the region R into smaller rectangles. Each subinterval in the x-direction will have a length of Δx = 0.2, and each subinterval in the y-direction will have a length of Δy = 0.4.
The vertices of each rectangle can be determined as follows:
Rectangle 1: (1, 2) - (1.2, 2.4)
Rectangle 2: (1.2, 2) - (1.4, 2.4)
Rectangle 3: (1.4, 2) - (1.6, 2.4)
Rectangle 4: (1, 2.4) - (1.2, 2.8)
Rectangle 5: (1.2, 2.4) - (1.4, 2.8)
Rectangle 6: (1.4, 2.4) - (1.6, 2.8)
Rectangle 7: (1, 2.8) - (1.2, 3.2)
Rectangle 8: (1.2, 2.8) - (1.4, 3.2)
Rectangle 9: (1.4, 2.8) - (1.6, 3.2)
Now, let's evaluate f(x, y) at a certain point within each rectangle. The given values in the table correspond to the midpoints of each rectangle, so we will use those values.
Riemann sum:
∫Rf(x, y)da ≈ Δx Δy [f(1.1, 2.2) + f(1.3, 2.2) + f(1.5, 2.2) + f(1.1, 2.6) + f(1.3, 2.6) + f(1.5, 2.6) + f(1.1, 3) + f(1.3, 3) + f(1.5, 3)]
Plugging in the corresponding values from the table:
∫Rf(x, y)da ≈ (0.2)(0.4)[-5 + 3 + 10 + 6 + (-5) + 2 + 6 + 8 + 10]
∫Rf(x, y)da ≈ (0.2)(0.4) [35]
∫Rf(x, y)da ≈ 0.08 [35]
∫Rf(x, y)da ≈ 2.8
Therefore, a reasonable estimate for ∫Rf(x, y)da is 2.8.
0.2*0.4 = 0.08 then u should get the right answer
this is just like the Riemann sum for f(x).
You have 9 rectangles, each of area 0.2 * 0.4 = 0.8
So add up 0.8 * f(x,y) at each of the given midpoints.