Upper and lower rectangles and numerical approximations.
Example 1
The curve y=2x2+2 is shown on [2,7]. Calculate the lower estimate L5 for the area under the curve, using 5 equal-width rectangles.
y
234567050100 x
y=2x2+2RectanglesShow worked answer
Answer: 190
- The curve is increasing on this interval.
- For a lower estimate, use the left endpoint of each subinterval, where the height is smallest.
- Each rectangle has width 57−2=1.
- L5=1×(10+20+34+52+74)=190.
Example 2
The curve y=2(3−x)2+4 is shown on [0,3]. Calculate the upper estimate U3 for the area under the curve, using 3 equal-width rectangles.
y
012301020 x
y=2(3−x)2+4RectanglesShow worked answer
Answer: 40
- The curve is decreasing on this interval.
- For a upper estimate, use the left endpoint of each subinterval, where the height is largest.
- Each rectangle has width 33−0=1.
- U3=1×(22+12+6)=40.
Example 3
The shaded rectangles estimate the area under y=x2+6 on [2,12]. Classify the estimate shown.
y
24681012050100150 x
y=x2+6RectanglesShow worked answer
Answer: A lower estimate
- The curve is increasing on this interval.
- For a lower estimate, use the left endpoint of each subinterval, where the height is smallest.
- Each rectangle lies below the curve within its subinterval, giving a lower estimate.