didn't we already do this?
A = 1/2 (f(2) + f(2.5) + ... + f(4.5))
The area is approximately=
A = 1/2 (f(2) + f(2.5) + ... + f(4.5))
and also the others, then get a final numeric answer.
Do the math!
First, calculate the width of each subinterval by dividing the total width of the interval by the number of subdivisions:
Width of each subinterval = (5 - 2) / 6 = 3 / 6 = 1/2
Next, calculate the left endpoint of each subinterval by starting at the left endpoint of the interval [2,5] and adding the width of each subinterval:
Left endpoints:
x1 = 2
x2 = 2 + (1/2) = 2.5
x3 = 2 + 2(1/2) = 3
x4 = 2 + 3(1/2) = 3.5
x5 = 2 + 4(1/2) = 4
x6 = 2 + 5(1/2) = 4.5
Now, calculate the area of each rectangle by evaluating the function at each left endpoint and multiplying it by the width of the subinterval:
Area of rectangle 1 = f(x1) * (width of subinterval) = (x1^2 + 1) * (1/2)
Area of rectangle 2 = f(x2) * (width of subinterval) = (x2^2 + 1) * (1/2)
Area of rectangle 3 = f(x3) * (width of subinterval) = (x3^2 + 1) * (1/2)
Area of rectangle 4 = f(x4) * (width of subinterval) = (x4^2 + 1) * (1/2)
Area of rectangle 5 = f(x5) * (width of subinterval) = (x5^2 + 1) * (1/2)
Area of rectangle 6 = f(x6) * (width of subinterval) = (x6^2 + 1) * (1/2)
Finally, calculate the sum of the areas of all the rectangles to get the approximate area under the curve:
Approximate area = Area of rectangle 1 + Area of rectangle 2 + Area of rectangle 3 + Area of rectangle 4 + Area of rectangle 5 + Area of rectangle 6