8. Definite Integrals

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 $f\left(x\right)=x^2$.

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

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?

           10           10

Suppose that Σ aₖ = -2 and Σ bₖ = 25. Find the value of

           k = 1          k = 1

  10

c. Σ (aₖ + bₖ - 1)

  k = 1

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 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.

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.

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

(d) Assuming the velocity remains 10 m/s, for t ≥ 5, find the function that gives the displacement between t = 0 and any time t ≥ 5.

Mass from density A thin 10-cm rod is made of an alloy whose density varies along its length according to the function shown in the figure. Assume density is measured in units of g/cm. In Chapter 6, we show that the mass of the rod is the area under the density curve.

(a) Find the mass of the left half of the rod (0 ≤ x ≤ 5) .