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

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.

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?

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. Assume ƒ and ƒ' are continuous functions for all real numbers.

(d) If ƒ is continuous on [a,b] and ∫ₐᵇ |ƒ(𝓍)| d𝓍 = 0 , then ƒ(𝓍) = 0 on [a,b] .

Displacement from a velocity graph Consider the velocity function for an object moving along a line (see figure).

(a) Describe the motion of the object over the interval [0,6].

Displacement from a velocity graph Consider the velocity function for an object moving along a line (see figure).

(b) Use geometry to find the displacement of the object between t = 0 and t = 2.

Displacement from a velocity graph Consider the velocity function for an object moving along a line (see figure).

(c) Use geometry to find the displacement of the object between t = 2 and t = 5.