The Riemann sum approximations for sin 0° using left endpoints, right endpoints, and midpoints are 0.167, 0.186, and 0.176, respectively. The most accurate is the midpoint rule, 0.176.
To approximate sin 0° with a Riemann sum with 10 intervals using:
(a) left endpoints:
We divide the interval [0°, 10°] into 10 subintervals of equal width:
[0°, 1°]
[1°, 2°]
[2°, 3°]
...
[9°, 10°]
At the left endpoint of each subinterval, we evaluate the function sin x. These values are:
sin 0° = 0
sin 1° = 0.0175
sin 2° = 0.0349
...
sin 9° = 0.1564
We then multiply these values by the width of each subinterval (1°) to get the area of each rectangle. These areas are:
0 * 1° = 0
0.0175 * 1° = 0.0175
0.0349 * 1° = 0.0349
...
0.1564 * 1° = 0.1564
Finally, we sum the areas of all the rectangles to get the Riemann sum approximation:
0 + 0.0175 + 0.0349 + ... + 0.1564 = 0.167
Therefore, the Riemann sum approximation for sin 0° using left endpoints is 0.167.
(b) right endpoints:
We follow the same steps as in (a), but instead of evaluating the function at the left endpoint of each subinterval, we evaluate it at the right endpoint. These values are:
sin 1° = 0.0175
sin 2° = 0.0349
...
sin 10° = 0.1736
The corresponding areas of the rectangles are:
0.0175 * 1° = 0.0175
0.0349 * 1° = 0.0349
...
0.1736 * 1° = 0.1736
The Riemann sum approximation for sin 0° using right endpoints is:
0.0175 + 0.0349 + ... + 0.1736 = 0.186
Therefore, the Riemann sum approximation for sin 0° using right endpoints is 0.186.
(c) midpoints:
We follow the same steps as in (a) and (b), but instead of evaluating the function at the left or right endpoint of each subinterval, we evaluate it at the midpoint. These values are:
sin 0.5° = 0.0087
sin 1.5° = 0.0259
...
9.5° = 0.1673
The corresponding areas of the rectangles are:
0.0087 * 1° = 0.0087
0.0259 * 1° = 0.0259
.....
0.1673 * 1° = 0.1673
The Riemann sum approximation for sin 0° using midpoints is:
0.0087 + 0.0259 + ... + 0.1673 = 0.176
Therefore, the Riemann sum approximation for sin 0° using midpoints is 0.176.
Conclusion:
The Riemann sum approximations for sin 0° using left endpoints, right endpoints, and midpoints are 0.167, 0.186, and 0.176, respectively. The most accurate approximation is the midpoint rule, which is 0.176.