In the case of a decreasing, concave down parabola in the intervals 0 to 1 with all of the y values positive, would the actual area be less or more than the estimated area using the trapezoid rule? And would it be less or more than using the midpoint rule? I know the left endpoint rectangles would give the highest area and the right endpoints would give the lowest, but I'm not sure about the others.
take a look at
http://www.math.pitt.edu/~sparling/052/23052/23052notes/23052notestojan14th/node3.html
The trapezoid rule gives more accurate estimate of the actual area. For that curve, on the left side of max, it give too high, on the right side, to low . Those areas ought to balance out.
To determine whether the actual area would be less or more than the estimated area using the trapezoid rule and the midpoint rule for a decreasing, concave down parabola, you first need to understand how each of these methods estimates the area.
1. Trapezoid Rule:
The trapezoid rule estimates the area under a curve by dividing the region into trapezoids and summing up their individual areas. The rule approximates the curve with straight lines connecting consecutive points. The estimated area using the trapezoid rule tends to lie between the actual area and the area calculated using the left and right endpoints.
2. Midpoint Rule:
The midpoint rule estimates the area under a curve by dividing the region into rectangles and summing up their individual areas. Each rectangle's height is determined by the midpoint of its base. The estimated area using the midpoint rule tends to be closer to the actual area compared to the trapezoid rule.
Now, let's analyze the specific scenario you mentioned: a decreasing, concave down parabola in the intervals 0 to 1 with all y-values positive.
Since the parabola is concave down, the trapezoids used to approximate the curve will lie above the actual curve. This means that the estimated area using the trapezoid rule will be greater than the actual area in this case.
On the other hand, the midpoint rule uses rectangles, which are less likely to overshoot the actual curve. As a result, the estimated area using the midpoint rule will be more accurate and likely to be closer to the actual area compared to the trapezoid rule. However, it is still not guaranteed to be exactly equal to the actual area.
To summarize:
- The estimated area using the trapezoid rule will be greater than the actual area.
- The estimated area using the midpoint rule will be more accurate and likely to be closer to the actual area compared to the trapezoid rule.
It is worth noting that both methods are approximations and can still introduce some degree of error, especially when dealing with curvatures, like concave down parabolas.