Final answer:
To find new estimates using ten rectangles, we divide the interval into subintervals and calculate the area of a rectangle for each subinterval. We can find the lower estimate by taking the minimum height, the upper estimate by taking the maximum height, and the midpoint estimate by taking the average of the minimum and maximum heights. The trapezoidal rule involves dividing the interval into subintervals and calculating the area of a trapezoid for each subinterval.
Step-by-step explanation:
To find new estimates using ten rectangles, we can divide the interval into ten equal subintervals. For each subinterval, we can calculate the area of a rectangle. a) Lower estimate: To find the lower estimate, we take the minimum height of the function in each subinterval. Then, we multiply the height by the width of the subinterval. Finally, we add up the areas of all the rectangles. b) Upper estimate: To find the upper estimate, we take the maximum height of the function in each subinterval. Then, we multiply the height by the width of the subinterval. Finally, we add up the areas of all the rectangles. c) Midpoint estimate: To find the midpoint estimate, we take the average of the minimum and maximum heights of the function in each subinterval. Then, we multiply the average height by the width of the subinterval. Finally, we add up the areas of all the rectangles. d) Trapezoidal rule: To calculate the area under the curve using the trapezoidal rule, we divide the interval into equal subintervals. For each subinterval, we calculate the area of a trapezoid. To do this, we take the average of the heights of the function at the beginning and end of the subinterval. Then, we multiply the average height by the width of the subinterval. Finally, we add up the areas of all the trapezoids.