BackApproximating area from a graph Approximate the area of the region bounded by the graph (see figure) and the 𝓍-axis by dividing the interval [1, 7] into n = 6 subintervals. Use a left and right Riemann sum to obtain two different approximations.
Suppose the interval [1, 3] is partitioned into n = 4 subintervals. What is the subinterval length ∆𝓍? List the grid points x₀ , x₁ , x₂ , x₃ and x₄. Which points are used for the left, right, and midpoint Riemann sums?
Does a right Riemann sum underestimate or overestimate the area of the region under the graph of a function that is positive and decreasing on an interval [a,b]? Explain.
Approximating displacement The velocity in ft/s of an object moving along a line is given by v = 3t² + 1 on the interval 0 ≤ t ≤ 4, where t is measured in seconds.
(a) Divide the interval [0,4] into n = 4 subintervals, [0,1] , [1.2] , [2,3] , and [3,4]. On each subinterval, assume the object moves at a constant velocity equal to v evaluated at the midpoint of the subinterval, and use these approximations to estimate the displacement of the object on [0, 4] (see part (a) of the figure)
{Use of Tech} Riemann sums for larger values of n Complete the following steps for the given function f and interval.
ƒ(𝓍) = 3 √x on [0,4] ; n = 40
(b) Based on the approximations found in part (a), estimate the area of the region bounded by the graph of f and the x-axis on the interval.
{Use of Tech} Riemann sums for larger values of n Complete the following steps for the given function f and interval.
ƒ(𝓍) = x² ― 1 on [2,5] ; n = 75
(b) Based on the approximations found in part (a), estimate the area of the region bounded by the graph of f and the x-axis on the interval.
Approximating displacement The velocity of an object is given by the following functions on a specified interval. Approximate the displacement of the object on this interval by subdividing the interval into n subintervals. Use the left endpoint of each subinterval to compute the height of the rectangles.
v = 2t + 1(m/s), for 0 ≤ t ≤ 8 ; n = 2
Approximating displacement The velocity of an object is given by the following functions on a specified interval. Approximate the displacement of the object on this interval by subdividing the interval into n subintervals. Use the left endpoint of each subinterval to compute the height of the rectangles.
v = [1 / (2t + 1)] (m/s), for 0 ≤ t ≤ 8 ; n = 4
Approximating displacement The velocity of an object is given by the following functions on a specified interval. Approximate the displacement of the object on this interval by subdividing the interval into n subintervals. Use the left endpoint of each subinterval to compute the height of the rectangles.
{Use of Tech} v = 4 √(t +1) (mi/hr) . for 0 ≤ t ≤ 15 ; n = 5
Left and right Riemann sums Use the figures to calculate the left and right Riemann sums for f on the given interval and for the given value of n.
ƒ(𝓍) = x + 1 on [1,6] ; n = 5
Left and right Riemann sums Complete the following steps for the given function, interval, and value of n.
f(x) = x + 1 on [0,4]; n = 4
(d) Calculate the left and right Riemann sums.
Left and right Riemann sums Complete the following steps for the given function, interval, and value of n.
{Use of Tech} ƒ(𝓍) = cos 𝓍 on [0. π/2]; n = 4
(d) Calculate the left and right Riemann sums.
Left and right Riemann sums Complete the following steps for the given function, interval, and value of n.
ƒ(𝓍) = x² ─ 1 on [2,4]; n = 4
(d) Calculate the left and right Riemann sums.
Left and right Riemann sums Complete the following steps for the given function, interval, and value of n.
{Use of Tech} ƒ(𝓍) = e ˣ/₂ on [1,4]; n = 6
(d) Calculate the left and right Riemann sums.
A midpoint Riemann sum Approximate the area of the region bounded by the graph of ƒ(𝓍) = 100 ― x² and the x-axis on [0, 10] with n = 5 subintervals. Use the midpoint of each subinterval to determine the height of each rectangle (see figure).