Estimating area, Riemann sums and the trapezium rule
Estimate the area under y = x² with thin rectangles and with trapezia, and see which number the estimates head for as the strips get narrower.
- فلسطين: الصف ١٢
- إيران: الصف ١٢
- أفغانستان: الصف ١٢
- أوكرانيا: الصف ١١
- روسيا: الصف ١١
- ألمانيا: الصف ١٢
- بريطانيا: السنة ١٢
لم يُترجَم هذا المختبر إلى هذه اللغة بعد.
الدليل
The area under a curve cannot be found directly, but we can cut it into thin strips. We replace each strip by a rectangle: area = width × height. The height is read at the left end of the strip (left sum) or at its right end (right sum).
For an increasing function the left sum is smaller and the right sum is bigger than the true area. If we take each strip as a trapezium, its area is a + b٢×h, where a and b are the two side heights and h is the width of the strip.
The narrower the strips, the closer the estimates get to one number. That limit is the التكامل المحدد, and the sums that lead to it are مجموع ريمانs.