using 5 rectangles what is the area under a curve using the function f(x)=3x+4 with boundries at [0,2]
To calculate the area under the curve using rectangles, we can use the method of Riemann sums. In this case, we will approximate the area under the curve of the function f(x) = 3x + 4 between x = 0 and x = 2 using 5 rectangles.
First, we need to divide the interval [0, 2] into 5 equal subintervals. The width of each rectangle will be the width of each subinterval.
Width of each subinterval = (b - a) / n = (2 - 0) / 5 = 0.4
Next, we need to calculate the height of each rectangle by evaluating the function at the midpoint of each subinterval. The midpoint of each subinterval can be found by taking the average of its boundaries.
Midpoint of first subinterval = (0 + 0.4) / 2 = 0.2
Height of first rectangle = f(0.2) = 3(0.2) + 4 = 4.6
Midpoint of second subinterval = (0.4 + 0.8) / 2 = 0.6
Height of second rectangle = f(0.6) = 3(0.6) + 4 = 5.8
Midpoint of third subinterval = (0.8 + 1.2) / 2 = 1.0
Height of third rectangle = f(1.0) = 3(1.0) + 4 = 7.0
Midpoint of fourth subinterval = (1.2 + 1.6) / 2 = 1.4
Height of fourth rectangle = f(1.4) = 3(1.4) + 4 = 8.2
Midpoint of fifth subinterval = (1.6 + 2.0) / 2 = 1.8
Height of fifth rectangle = f(1.8) = 3(1.8) + 4 = 8.4
Now, we can calculate the area of each rectangle by multiplying its width and height.
Area of first rectangle = (0.4)(4.6) = 1.84
Area of second rectangle = (0.4)(5.8) = 2.32
Area of third rectangle = (0.4)(7.0) = 2.8
Area of fourth rectangle = (0.4)(8.2) = 3.28
Area of fifth rectangle = (0.4)(8.4) = 3.36
Finally, we sum up the areas of all the rectangles to get the approximate area under the curve.
Total area = sum of areas of all rectangles = 1.84 + 2.32 + 2.8 + 3.28 + 3.36 = 13.6
Therefore, the approximate area under the curve of f(x) = 3x + 4 between x = 0 and x = 2, using 5 rectangles, is 13.6.