Question

How do I find out which rule produced which estimate and the two approximations that the true value of `int_0^2` f(x) dx lie in?

The left, right, Trapezoidal, and Midpoint Rule approximations were used to estimate `int_0^2` f(x) dx, where f is the function whose graph is shown. The estimates were `color(red)(0.7819)`, `color(red)(0.8689)`, `color(red)(0.8632)`, and `color(red)(0.9560)`, and the same number of subintervals were used in each case.

(a) Which rule produced which estimate?
`L_n` = ???
`R_n` = ???
`T_n` = ???
`M_n` = ???

(b) Between which two approximations does the true value of `int_0^2` f(x) dx lie?

Smaller value = ???
Larger value = ???

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Answer 1

`L_n = 0.9560`
`R_n = 0.7819`
`T_n = 0.8689`
`M_n = 0.8632`

The true value of the integral lies between `M_n = 0.8632` (smaller) and `T_n = 0.8689` (bigger).

Explanation:

This function is decreasing and concave up. Therefore, the following statements are true:

The right sum will be the least of the four sums, since the values used for the bars will be as low as possible (since the function is decreasing to the right). This can be seen in the image below:

The left sum will be the most of the four sums, since the values used for the bars will be as high as possible. This can be seen in the image below:

The trapezoidal sum will be slightly more than the actual value of the integral, since the function is concave up. The reason for this is that the trapezoidal sum connects the left and right endpoints of each interval with a straight line segment, while the actual curve dips down below this line segment and then meets back up with it at the end of the interval. This can be seen in the image below (although it's not as noticeable as with the right and left sums):

Finally, the midpoint sum will be slightly less than the actual value of the integral, since the function is concave up. The reason for this is that (since the function is concave up) the function decreases the fastest at the beginning of each interval compared to the end, so the midpoint of each interval is actually below the average of the two endpoints. This can (sort of) be seen in the graph below (although it isn't perfect):

Based on these four descriptions, we can determine that, from least to greatest, the four sums are:

Right, Midpoint, Trapezoidal, Left

And since our values from least to greatest are:

0.7819, 0.8632, 0.8689, 0.9560

This means that:

Right sum `= 0.7819`
Midpoint sum `= 0.8632`
Trapezoidal sum `= 0.8689`
Left sum `= 0.9560`

So this is the answer for part A. As for part B, we can see from part A's answer that the true value of the integral lies between:

The smaller value: Midpoint sum (0.8632)

The larger value: Trapezoidal sum (0.8689)

Final Answer