Use three rectangles to estimate the area under the curve of $f\left(x\right)=\frac{1}{3}{x}^3$ from $x=0$ to $x=3$ using the right endpoints.

The velocity in ft/s of an object moving along a line is given by v = ƒ(t) on the interval 0 ≤ t ≤ 6 (see figure), where t is measured in seconds.

(a) Divide the interval [0,6] into n = 3 subintervals, [0,2] , [2,4] and [4,6]. On each subinterval, assume the object moves at a constant velocity equal to the value of v evaluated at the right endpoint of the subinterval, and use these approximations to estimate the displacement of the object on [0,6] (see part (a) of the figure)                                                                                                             

Use five rectangles to estimate the area under the curve of $f\left(x\right)=x^2$ from $x=0$ to $x=5$ using left endpoints.

Use two rectangles to estimate the area under the curve of $f\left(x\right)=\frac{1}{2}{x}^2$ from $x=0$ to $x=3$ using left endpoints.

Use four rectangles to estimate the area under the curve of $f\left(x\right)=x^2+2$ from $x=0$ to $x=2$ using left endpoints.

Use three rectangles to approximate the area under the curve of $f\left(x\right)=3{(x-2)}^2$ from $x=0$ to $x=3$ using the midpoint rule.

Estimate the value of the definite integral ${\int }_0^{10}f\left(x\right)dx$ using five subintervals and the left endpoint approximation, given that $f\left(x\right)=x^2$.

Estimate the value of the definite integral ${\int }_{-2}^{4}g\left(x\right)dx$ using six subintervals and the left endpoint approximation. Which of the following best describes the process?

Evaluate the integral by interpreting it in terms of areas: $6{\int }_{-5}^0\left|x\right| dx$.