Porisma

A hint to get started

Take the limit of left-rectangle areas

The diagram shows part of the graph of y=x3+1y=x^3+1 from x=0x=0 to x=1x=1. The area under the curve may be approximated by the total area, AA, of nn rectangles as shown. The width of each rectangle is 1/n1/n.

Given that ∑r=1nr3=[n(n+1)2]2\displaystyle\sum_{r=1}^{n}r^3=\left[\frac{n(n+1)}2\right]^2, show that A=1+14(1−1n)2A=1+\dfrac14\left(1-\dfrac1n\right)^2.

Hence deduce the exact value of ∫01(x3+1) dx\displaystyle\int_0^1(x^3+1)\,dx.

Additional printed Student lecture ask: Construct a similar diagram so that overestimation occurs.

Expand diagram

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Increasing curve y=x³+1 on 0≤x≤1. Eight equal-width left rectangles illustrate the source construction for arbitrary n; each width is 1/n, first height 1, last height ((n-1)/n)³+1. Rectangles lie below the curve, with 0,1/n,2/n,(n-1)/n,1 marked on x-axis.
Increasing curve y=x³+1 on 0≤x≤1. Eight equal-width left rectangles illustrate the source construction for arbitrary n; each width is 1/n, first height 1, last height ((n-1)/n)³+1. Rectangles lie below the curve, with 0,1/n,2/n,(n-1)/n,1 marked on x-axis.
Increasing curve y=x³+1 on 0≤x≤1. Eight equal-width left rectangles illustrate the source construction for arbitrary n; each width is 1/n, first height 1, last height ((n-1)/n)³+1. Rectangles lie below the curve, with 0,1/n,2/n,(n-1)/n,1 marked on x-axis.

[2008/TPJC/II/5(modified)]

CJC · Discussion · Question 1

Read the rectangle heights from their left endpoints: the last input is (n−1)/n, not 1.

The repeated Student lecture copy adds an overestimation construction; use right endpoints for this increasing curve. The Teacher copy instead asks students to explain the given underestimate.

Read the worked solution