Writing Integrals Using Sigma Notation

 $\int_2^5\left(x^3+3\right)dx$  as limit of a sum is equivalent to

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

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

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

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

 $\int_0^{\pi}\cos xdx\$  as limit of a sum is equivalent to

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

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

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

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

 $\lim_{n\rightarrow\infty}\sum_{i=1}^n\left[\left(\frac{5i}{n}\right)^2+\frac{5i}{n}+1\right]\frac{5}{n}$  in integral notation would be 

•  $\int_0^5\left(x^2+x+1\right)dx$  

•  $\int_5^6\left(x^2+x+1\right)dx$  

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

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

 $\lim_{n\rightarrow\infty}\sum_{i=1}^n\left[2+\sqrt{3+\frac{4i}{n}}\right]\frac{4}{n}$  in integral notation would be 

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

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

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

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

Write the limit as a definite integral.

 $$   $\lim_{n\rightarrow\infty}\ \frac{\frac{\pi}{2}}{n}\sum_{k=1}^n\sec\left(k\cdot\frac{\pi}{2n}\right)$  

•  $\int_0^{\frac{\pi}{2}}\sec\left(x\right)dx$  

•  $\int_0^1\sec\left(x\right)dx$  

•  $\int_0^{\frac{\pi}{2}}\sec\left(\frac{1}{x}\right)dx$  

•  $\int_0^{\frac{\pi}{2}}\sec\left(\pi x\right)dx$  

 $\lim_{n\rightarrow\infty}\ \frac{1}{n}\sum_{k=1}^n\frac{1}{\sqrt{\frac{k}{n}}}$  Which answer choice is the definite integral for this limit?

•  $\int_0^1\sqrt{\frac{1}{x}}dx$  

•  $\int_0^1\sqrt{x}dx$  

•  $\int_0^{\sqrt{x}}\ \frac{1}{x}dx$  

•  $\int_0^1\ \frac{1}{\sqrt{x}}dx$