Calculate the left and right Riemann sums for ${\int }_1^4(2-x)dx$ with $n=6$ subintervals.

Calculate the left and right Riemann sums for ${\int }_2^8\frac{6}{x}dx$ using $n=6$ subintervals.

Use the definition of the definite integral with right Riemann sums to evaluate ${\int }_2^5(x^2+2)dx$.

Express the following limit of Riemann sums as a definite integral by identifying the function:

$\underset{\Delta \to 0}{\lim }\sum _{k=1}^n(5-x_k^{\ast })\Delta x_k\text{ on }[2,6]$

A particle's velocity is given by $v=\frac{3}{t+4}$ ($m/s$) for $2\le t\le 8$. Approximate the displacement of the particle over this interval by dividing it into $n=3$ subintervals and using the left endpoint of each subinterval for the rectangle heights.