Verified step by step guidance
05:43
06:21
04:27Area functions for the same linear function Let Ζ(t) = t and consider the two area functions A(π) = β«βΛ£ Ζ(t) dt and F(π) = β«βΛ£ Ζ(t) dt .
(b) Evaluate F(4) and F(6). Then use geometry to find an expression for F (π) , for π β₯ 2.
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
(a) If Ζ is symmetric about the line π = 2 , then β«ββ΄ Ζ(π) dπ = 2 β«βΒ² Ζ(π) dπ.
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.
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 β€ x β€ 5) .
Approximating areas Estimate the area of the region bounded by the graph of Ζ(π) = xΒ² + 2 and the x-axis on [0, 2] in the following ways.
(a) Divide [0, 2] into n = 4 subintervals and approximate the area of the region using a left Riemann sum. Illustrate the solution geometrically.
{Use of Tech} Approximating net area The following functions are positive and negative on the given interval.
f(π) = xΒ³ on [-1,2]
(b) Approximate the net area bounded by the graph of f and the x-axis on the interval using a left, right, and midpoint Riemann sum with n = 4.
