Riemann's Sum and Summation Notation (11/10): Check for Understanding

$$\lim_{n\rightarrow\infty}\sum_{i=1}^n\left[\left(2+\frac{3i}{n}\right)^3+3\right]\frac{1}{n}$$  

$$\lim_{n\rightarrow\infty}\sum_{i=1}^n\left[\left(2+\frac{3i}{n}\right)^3+3\right]\frac{3i}{n}$$  

$$\lim_{n\rightarrow\infty}\sum_{i=1}^n\left[\left(\frac{3i}{n}\right)^3+3\right]\frac{3}{n}$$  

$$\lim_{n\rightarrow\infty}\sum_{i=1}^n\left[\left(2+\frac{3i}{n}\right)^3+3\right]\frac{3}{n}$$  

$$\lim_{n\rightarrow\infty}\sum_{i=1}^n\left[\cos\left(\frac{\pi i}{n}\right)\right]\frac{i}{n}$$  

$$\lim_{n\rightarrow\infty}\sum_{i=1}^n\left[\cos\left(\frac{i}{n}\right)\right]\frac{i}{n}$$  

$$\lim_{n\rightarrow\infty}\sum_{i=1}^n\left[\cos\left(\frac{\pi i}{n}\right)\right]\frac{\pi}{n}$$  

$$\lim_{n\rightarrow\infty}\sum_{i=1}^n\left[\cos\left(\frac{i}{n}\right)\right]\frac{\pi}{n}$$  

$$\int_0^5\left(x^2+x+1\right)dx$$  

$$\int_5^6\left(x^2+x+1\right)dx$$  

$$\int_0^1\left(\left(5x\right)^2+5x+1\right)dx$$  

$$\int_0^{10}\left(\frac{x^2}{2}+\frac{x}{2}+1\right)dx$$  

$$\int_3^7\left(2+\sqrt{x}\right)dx$$  

$$\int_0^4\left(2+\sqrt{x}\right)dx$$  

$$\int_3^7\left(2x+\sqrt{x}\right)dx$$  

$$\int_3^7\left(2+\sqrt{3+x}\right)dx$$  

Left endpoint:  $$\frac{1}{2}\sum_{n=1}^6\left(4+\sqrt{\left(0.5n+5\right)-3}\right)\approx17.771$$  

Right endpoint: $$\frac{1}{2}\sum_{n=0}^5\left(4+\sqrt{\left(0.5n+5\right)-3}\right)\approx17.360$$  

Left endpoint:  $$\frac{1}{2}\sum_{n=0}^5\left(4+\sqrt{\left(0.5n\right)-3}\right)\approx17.360$$  
Right endpoint:  $$\frac{1}{2}\sum_{n=1}^6\left(4+\sqrt{\left(0.5n\right)-3}\right)\approx17.771$$  

Left endpoint:  $$\frac{1}{2}\sum_{n=0}^5\left(4+\sqrt{\left(0.5n+5\right)-3}\right)\approx17.360$$

Right endpoint:  $$\frac{1}{2}\sum_{n=1}^6\left(4+\sqrt{\left(0.5n+5\right)-3}\right)\approx17.771$$  

Left endpoint:  $$\frac{1}{2}\sum_{n=1}^6\left(4+\sqrt{\left(0.5n\right)-3}\right)\approx17.360$$  
Right endpoint:  $$\frac{1}{2}\sum_{n=0}^5\left(4+\sqrt{\left(0.5n\right)-3}\right)\approx17.771$$  

4 intervals, with heights using middle values

4 intervals, with heights using left values

5 intervals, with heights using middle values

5 intervals, with heights using right values

15,475

15,750

15,200

17,100

6

10

11

20