Approximate the area under the curve y = x^3 + 1 from x = 0 to x = 3 using a left sum with 6 subdivisions.
The area is approximately?
so, each rectangle has width 1/2. That makes the area
1/2 (f(0) + f(1/2) + ... + f(5/2))
So plug and chug,
There are several handy Riemann sum calculators online for you to check your work.
To approximate the area under the curve using a left Riemann sum, we will divide the interval [0, 3] into 6 equal subdivisions.
First, we need to calculate the width of each subdivision:
Width of each subdivision = (upper limit - lower limit) / number of subdivisions
= (3 - 0) / 6
= 3/6
= 1/2
Now, we will calculate the left Riemann sum by evaluating the function at the left endpoints of each subdivision and summing the results.
Subdivision 1: x = 0
f(x) = (0)^3 + 1 = 1
Area of Subdivision 1 = width x f(x) = (1/2) x 1 = 1/2
Subdivision 2: x = 1/2
f(x) = (1/2)^3 + 1 = 1/8 + 1 = 9/8
Area of Subdivision 2 = width x f(x) = (1/2) x (9/8) = 9/16
Subdivision 3: x = 1
f(x) = (1)^3 + 1 = 1 + 1 = 2
Area of Subdivision 3 = width x f(x) = (1/2) x 2 = 1
Subdivision 4: x = 3/2
f(x) = (3/2)^3 + 1 = 27/8 + 1 = 35/8
Area of Subdivision 4 = width x f(x) = (1/2) x (35/8) = 35/16
Subdivision 5: x = 2
f(x) = (2)^3 + 1 = 8 + 1 = 9
Area of Subdivision 5 = width x f(x) = (1/2) x 9 = 9/2
Subdivision 6: x = 5/2
f(x) = (5/2)^3 + 1 = 125/8 + 1 = 133/8
Area of Subdivision 6 = width x f(x) = (1/2) x (133/8) = 133/32
Now, we add up the areas of all the subdivisions to get the approximate area under the curve:
Approximate area = (1/2) + (9/16) + 1 + (35/16) + (9/2) + (133/32)
Evaluating the sum, the approximate area under the curve is:
Approximate area = 165/16 or 10.3125
To approximate the area under the curve using a left sum, we can use the following steps:
1. Determine the width of each subdivision. In this case, we have 6 subdivisions over the interval from x = 0 to x = 3. Therefore, the width of each subdivision is (3 - 0) / 6 = 0.5.
2. Compute the left sum by evaluating the function at the left endpoint of each subdivision, and then summing up the areas of the rectangles formed.
Let's calculate the left sum:
For the first subdivision (from x = 0 to x = 0.5):
Evaluate the function at the left endpoint (x = 0) and multiply it by the width.
Area of first rectangle = f(0) * 0.5 = (0^3 + 1) * 0.5 = 1 * 0.5 = 0.5.
Repeat the process for the remaining subdivisions, and sum up the areas obtained:
For the second subdivision (from x = 0.5 to x = 1):
Area of second rectangle = f(0.5) * 0.5 = (0.5^3 + 1) * 0.5.
For the third subdivision (from x = 1 to x = 1.5):
Area of third rectangle = f(1) * 0.5 = (1^3 + 1) * 0.5.
Continue this process until the last subdivision (from x = 2.5 to x = 3):
Area of last rectangle = f(2.5) * 0.5 = (2.5^3 + 1) * 0.5.
Finally, sum up the areas of all the rectangles calculated to get the approximation for the area under the curve.
Area = Area of first rectangle + Area of second rectangle + ... + Area of last rectangle.
Now you can apply this method and calculate the approximation for the area under the curve y = x^3 + 1 from x = 0 to x = 3 using a left sum with 6 subdivisions.
well, I am not going to go through the pain of approximation with rectangles but your result should be something like the integral which is x^4/4 + x at 3 minus at 0
(81 / 4) + 3 - 0 = 23.25