Estimate the area under the curve f(x) = x2 + 1 from x = 0 to x = 6 by using three circumscribed (over the curve) rectangles. Answer to the nearest integer.
a. I assume under the curve means from f(x)=0 to the curve, and I assume circumstribed means right hand rectangle corner on the curve.
area= (x^2+1)at x=2,4,6
area=5+17+37
I'm intrigued how you got all those odd values for the areas when the width of each rectangle is 2 ....
georaphy
Steve you plug the x value into the y formula which causes you to add 1 to the even value... always check your work.
To estimate the area under the curve of f(x) = x^2 + 1 from x = 0 to x = 6 using three circumscribed (over the curve) rectangles, we can use the method of Riemann sums.
First, let's divide the interval [0, 6] into three equal subintervals. The width of each rectangle will be 6/3 = 2.
Next, we calculate the height of each rectangle by evaluating the function at the right endpoint of each subinterval.
For the first rectangle, the right endpoint is x = 2. So, the height of the first rectangle is f(2) = (2^2) + 1 = 5.
For the second rectangle, the right endpoint is x = 4. So, the height of the second rectangle is f(4) = (4^2) + 1 = 17.
For the third rectangle, the right endpoint is x = 6. So, the height of the third rectangle is f(6) = (6^2) + 1 = 37.
Now, we can calculate the area of each rectangle by multiplying its width by its height.
The area of the first rectangle = 2 * 5 = 10.
The area of the second rectangle = 2 * 17 = 34.
The area of the third rectangle = 2 * 37 = 74.
Finally, we add up the areas of all three rectangles to estimate the total area under the curve:
Total area ≈ 10 + 34 + 74 = 118.
Therefore, the estimated area under the curve f(x) = x^2 + 1 from x = 0 to x = 6, using three circumscribed rectangles, is approximately 118 (to the nearest integer).