For the function whose values are given in the table below, the integral from 0 to six of f(x)dx is approximated by a Reimann Sum using the value at the LEFT ENDPOINT value of each of three intervals with width 2.
X 0 1 2 3 4 5 6
f(x) 0 0.25 0.48 0.68 0.84 0.95 1
The Approximation is:
a) 2.64
b) 3.64
c) 3.72
d) 3.76
you would add 0+0.48+0.84 and then multiple that by 2. Its those numbers because its asking for the left endpoint. If it were asking for the right endpoint, it would be 0.48+0.84+1 and multiplied by 2. Its multiplied by 2 because that is the width of the three triangles
=a. 2.64 for left endpoint
To approximate the integral using a Riemann Sum with left endpoints, we need to divide the interval [0, 6] into three subintervals, each with a width of 2.
First, let's calculate the height of each rectangle by using the left endpoint values of f(x):
Height of first rectangle = f(0) = 0
Height of second rectangle = f(2) = 0.48
Height of third rectangle = f(4) = 0.84
Next, calculate the area of each rectangle by multiplying the height by the width (2):
Area of first rectangle = 0 * 2 = 0
Area of second rectangle = 0.48 * 2 = 0.96
Area of third rectangle = 0.84 * 2 = 1.68
Finally, add up the areas of all the rectangles to get the approximation of the integral:
Approximation = Area of first rectangle + Area of second rectangle + Area of third rectangle
= 0 + 0.96 + 1.68
= 2.64
Therefore, the approximate value of the integral from 0 to 6 of f(x)dx using the left endpoint values is 2.64.
So, the answer is a) 2.64.
To approximate the integral using a Riemann sum with the left endpoint value of each interval, we need to calculate the sum of the areas of the rectangles formed by the function values.
First, let's calculate the width of each interval. In this case, it is given as 2.
Next, we will calculate the height of each rectangle, which is the value of the function at the left endpoint of each interval. The left endpoints for the three intervals are 0, 2, and 4. The corresponding function values are 0, 0.48, and 0.84, respectively.
Now, let's calculate the area of each rectangle. The area of a rectangle is given by multiplying its width by its height.
For the first rectangle, with a width of 2 and a height of 0, the area is 2 * 0 = 0.
For the second rectangle, with a width of 2 and a height of 0.48, the area is 2 * 0.48 = 0.96.
For the third rectangle, with a width of 2 and a height of 0.84, the area is 2 * 0.84 = 1.68.
Finally, we add up the areas of all the rectangles to get the approximation of the integral:
Approximation = 0 + 0.96 + 1.68 = 2.64.
Therefore, the approximation of the integral from 0 to 6 of f(x)dx using a Riemann sum with the left endpoint values of each interval with width 2 is 2.64.
The correct option is a) 2.64.