Answer to Question #270751 in Calculus for Thabza

Question

Suppose an area 𝑆 of a parabola 𝑦=𝑥2 illustrated in Figure 1.1 below is divided into four stripes 𝑆1,𝑆2,𝑆3, and 𝑆4 as shown in Figure 1.2 below; State the formulas to estimate the sum of the rectangles (𝑅1.3 and 𝑅1.4) under the parabolic region 𝑆 illustrated in Figures 1.3, 1.4 and the sum of areas of 𝑝 rectangles.

Answer 1

Let $S_n$ be the sum of the areas of the rectangles. Each rectangle has width $1/n$ and the heights are the values of the function $f(x)=x^2$ at the points $1/n, 2/n, 3/n, ..., n/n.$ ; That is, the heights are $(1/n)^2, (2/n)^2,(3/n)^2, ..., (n/n)^2.$ Thus

$$S_3=\dfrac{1}{3}(\dfrac{1}{3})^2+\dfrac{1}{3}(\dfrac{2}{3})^2+\dfrac{1}{3}(\dfrac{3}{3})^2$$

$$=\dfrac{1}{3^3}(1^2+2^2+3^2)=\dfrac{14}{27}$$

$$S_4=\dfrac{1}{4}(\dfrac{1}{4})^2+\dfrac{1}{4}(\dfrac{2}{4})^2+\dfrac{1}{4}(\dfrac{3}{4})^2+\dfrac{1}{4}(\dfrac{4}{4})^2$$

$$=\dfrac{1}{4^3}(1^2+2^2+3^2+4^2)=\dfrac{15}{32}$$

The sum of areas of $p$ rectangles

$$S_p=\dfrac{1}{p}(\dfrac{1}{p})^2+\dfrac{1}{p}(\dfrac{2}{p})^2+...+\dfrac{1}{p}(\dfrac{p}{p})^2$$

$$=\dfrac{1}{p^3}\displaystyle\sum_{i=1}^pi^2=\dfrac{1}{p^3}(\dfrac{p(p+1)(2p+1)}{6})$$

$$=\dfrac{(p+1)(2p+1)}{6p^2}$$