BackA 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).
Sigma notation Express the following sums using sigma notation. (Answers are not unique.)
(a) 1 + 2 + 3 + 4 + 5
66–71. {Use of Tech} Estimating error Refer to Theorem 8.1 in the following exercises.
67. Let f(x) = √(x³ + 1).
d. Use Theorem 8.1 to find an upper bound on the absolute error in the estimate found in part (a).
37-40. {Use of Tech} Temperature data
Howdy temperature data for Boulder, Colorado; San Francisco, California; Nantucket, Massachusetts; and Duluth, Minnesota, over a 12-hr period on the same day of January are shown in the figure.
Assume these data are taken from a continuous temperature function T(t). The average temperature (in °F) over the 12-hr period is:
T_avg = (1/12) × ∫(0 to 12) T(t) dt
38. Find an accurate approximation to the average temperature over the 12-hr period for San Francisco. State your method.
The instructions for the integrals in Exercises 1–10 have three parts, one for the Midpoint Rule, one for the Trapezoidal Rule, and one for Simpson’s Rule.
III. Using Simpson's Rule
a. Estimate the integral with n = 4 steps and find an upper bound for |ES|.
∫ from 1 to 2 of 1 / s² ds
Evaluate the following summation:
In Exercises 11–22, estimate the minimum number of subintervals needed to approximate the integrals with an error of magnitude less than 10^-4 by (a) the Trapezoidal Rule (The integrals in Exercises 11–18 are the integrals from Exercises 1–8.)
∫ from 1 to 2 of 1/s² ds
Write the Riemann sum that would approximate the area of the following graph over the interval [0,3] using 3 subintervals.
[Technology Exercise] When solving Exercises 33-40, you may need to use a calculator or a computer.
Use numerical integration to estimate the value of
arcsin(0.6) = ∫ (from 0 to 0.6) dx / √(1 - x²).
For reference, arcsin(0.6) = 0.64350 to five decimal places.
{Use of Tech} Approximating definite integrals Complete the following steps for the given integral and the given value of n.
(d) Determine which Riemann sum (left or right) underestimates the value of the definite integral and which overestimates the value of the definite integral..
∫₀² (𝓍²―2) d𝓍 ; 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
{Use of Tech} Approximating definite integrals Complete the following steps for the given integral and the given value of n.
(c) Calculate the left and right Riemann sums for the given value of n.
∫₀² (𝓍²―2) d𝓍 ; n = 4
A piece of wood paneling must be cut in the shape shown in the figure.
The coordinates of several points on its curved surface are also shown (with units of inches).
a. Estimate the surface area of the paneling using the Trapezoid Rule.
{Use of Tech} Approximating definite integrals with a calculator Consider the following definite integrals.
(b) Evaluate each sum using a calculator with n = 20, 50, and 100. Use these values to estimate the value of the integral.
∫₀¹ cos ⁻¹ 𝓍 d𝓍
The instructions for the integrals in Exercises 1–10 have three parts, one for the Midpoint Rule, one for the Trapezoidal Rule, and one for Simpson’s Rule.
III. Using Simpson's Rule
b. Evaluate the integral directly and find |ES|.
∫ from 1 to 2 of x dx