Approximate the area under the graph of f(x)and above the x-axis using 4 rectangles: f(x)=x(x) +2 using 4 rectangles at intervals 0,5
To approximate the area under the graph of the function f(x) = x^2 + 2 and above the x-axis using 4 rectangles, you can use the method known as the "midpoint rule." The midpoint rule divides the area under the curve into rectangles of equal width and approximates the height of each rectangle by evaluating the function at the midpoint of each interval.
First, we need to determine the width of each rectangle. Since we are using 4 rectangles to cover the interval from x = 0 to x = 5, the width of each rectangle can be calculated by dividing the width of the interval (5 - 0) by the number of rectangles, which is 4. So each rectangle will have a width of 5/4 = 1.25.
Next, we need to determine the height of each rectangle. To do this, we will evaluate the function at the midpoint of each rectangle. Since we have 4 rectangles, we will have 5 intervals (the endpoints and the midpoints).
Interval 1: x = 0.625
f(0.625) = (0.625)^2 + 2 = 0.390625 + 2 = 2.390625
Interval 2: x = 1.875
f(1.875) = (1.875)^2 + 2 = 3.515625 + 2 = 5.515625
Interval 3: x = 3.125
f(3.125) = (3.125)^2 + 2 = 9.765625 + 2 = 11.765625
Interval 4: x = 4.375
f(4.375) = (4.375)^2 + 2 = 19.140625 + 2 = 21.140625
Interval 5: x = 5.625
f(5.625) = (5.625)^2 + 2 = 31.640625 + 2 = 33.640625
Now, we can calculate the area of each rectangle by multiplying the width by the height:
Rectangle 1: width = 1.25, height = 2.390625
Area = 1.25 * 2.390625 = 2.98828125
Rectangle 2: width = 1.25, height = 5.515625
Area = 1.25 * 5.515625 = 6.89453125
Rectangle 3: width = 1.25, height = 11.765625
Area = 1.25 * 11.765625 = 14.70703125
Rectangle 4: width = 1.25, height = 21.140625
Area = 1.25 * 21.140625 = 26.42578125
Finally, to approximate the total area under the curve, we sum the areas of all the rectangles:
Total Area = 2.98828125 + 6.89453125 + 14.70703125 + 26.42578125 = 51.015625
Therefore, the approximate area under the graph of f(x) = x^2 + 2 and above the x-axis using 4 rectangles is 51.015625 square units.