Approximate the area under the curve y = x^2 + 1 from x = 2 to x = 5 using a left sum with 6 subdivisions.
The area is approximately=
didn't we already do this?
A = 1/2 (f(2) + f(2.5) + ... + f(4.5))
its wrong when i put it in, the answer supposed to be a number
To approximate the area under the curve using a left sum with 6 subdivisions, you need to divide the interval [2, 5] into 6 equal subintervals and calculate the sum of the areas of the rectangles formed by the left endpoints of each subinterval and the curve.
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
oh come on. Evaluate f(2) = 5
and also the others, then get a final numeric answer.
Do the math!