To estimate the area under the curve y=x+1 between x=0 and x=10 with n=5 sub-intervals, you can use the following steps in a spreadsheet program like Microsoft Excel:
Create a table with three columns labeled "x", "y", and "Area".
In the "x" column, enter the values 0, 2, 4, 6, 8, and 10 (these are the endpoints and midpoints of the sub-intervals).
In the "y" column, enter the corresponding y-values for each x-value. For example, for x=0, y=1; for x=2, y=3; for x=4, y=5; and so on.
In the "Area" column, calculate the area of each sub-interval using the trapezoidal rule. The formula for the area of a trapezoid is:
Area = (base1 + base2) * height / 2
where base1 and base2 are the lengths of the two parallel sides of the trapezoid (which are the y-values of the function at the endpoints of the sub-interval), and height is the width of the sub-interval (which is 2 in this case).
Add up the areas of all the sub-intervals to get an estimate of the total area under the curve.
The completed table should look like this:
x y Area
0 1 0
2 3 4
4 5 8
6 7 8
8 9 4
10 11 0
To calculate the total area under the curve, simply sum the values in the "Area" column, which gives an estimate of 24 square units. This can be used to determine the amount of mulch needed for the garden bed.