Estimating the Number of SubintervalsEstimate

Problem 40E Effects of an antihistamine The concentration of an antihistamine in the bloodstream of a healthy adult is modeled by where C is measured in grams per liter and it is the time in hours since the medication was taken. What is the average level of con-centration in the bloodstream over a 6-hr period?

Problem 5E Estimating Integrals The instructions for the integrals in Exercises 1–10 have two parts, one for the Trapezoidal Rule and one for Simpson’s Rule. I. Using the Trapezoidal Rule a. Estimate the integral with n = 4 steps and find an upper bound for b. Evaluate the integral directly and find c. Use the formula ( /( true values)) × 100 to express as a percentage of the integral’s true value. II. Using Simpson’s Rule a. Estimate the integral with n = 4 steps and find an upper bound for b. Evaluate the integral directly and find c. Use the formula ( /( true values)) × 100 to express as a percentage of the integral’s true value.

Problem 2E Estimating Integrals The instructions for the integrals in Exercises 1–10 have two parts, one for the Trapezoidal Rule and one for Simpson’s Rule. I. Using the Trapezoidal Rule a. Estimate the integral with n = 4 steps and find an upper bound for b. Evaluate the integral directly and find c. Use the formula ( /( true values)) × 100 to express as a percentage of the integral’s true value. II. Using Simpson’s Rule a. Estimate the integral with n = 4 steps and find an upper bound for b. Evaluate the integral directly and find c. Use the formula ( /( true values)) × 100 to express as a percentage of the integral’s true value.

Problem 3E Estimating Integrals The instructions for the integrals in Exercises 1–10 have two parts, one for the Trapezoidal Rule and one for Simpson’s Rule. I. Using the Trapezoidal Rule a. Estimate the integral with n = 4 steps and find an upper bound for b. Evaluate the integral directly and find c. Use the formula ( /( true values)) × 100 to express as a percentage of the integral’s true value. II. Using Simpson’s Rule a. Estimate the integral with n = 4 steps and find an upper bound for b. Evaluate the integral directly and find c. Use the formula ( /( true values)) × 100 to express as a percentage of the integral’s true value.

Problem 4E Estimating Integrals The instructions for the integrals in Exercises 1–10 have two parts, one for the Trapezoidal Rule and one for Simpson’s Rule. I. Using the Trapezoidal Rule a. Estimate the integral with n = 4 steps and find an upper bound for b. Evaluate the integral directly and find c. Use the formula ( /( true values)) × 100 to express as a percentage of the integral’s true value. II. Using Simpson’s Rule a. Estimate the integral with n = 4 steps and find an upper bound for b. Evaluate the integral directly and find c. Use the formula ( /( true values)) × 100 to express as a percentage of the integral’s true value.

Problem 6E Estimating Integrals The instructions for the integrals in Exercises 1–10 have two parts, one for the Trapezoidal Rule and one for Simpson’s Rule. I. Using the Trapezoidal Rule a. Estimate the integral with n = 4 steps and find an upper bound for b. Evaluate the integral directly and find c. Use the formula ( /( true values)) × 100 to express as a percentage of the integral’s true value. II. Using Simpson’s Rule a. Estimate the integral with n = 4 steps and find an upper bound for b. Evaluate the integral directly and find c. Use the formula ( /( true values)) × 100 to express as a percentage of the integral’s true value.

Problem 7E Estimating Integrals The instructions for the integrals in Exercises 1–10 have two parts, one for the Trapezoidal Rule and one for Simpson’s Rule. I. Using the Trapezoidal Rule a. Estimate the integral with n = 4 steps and find an upper bound for b. Evaluate the integral directly and find c. Use the formula ( /( true values)) × 100 to express as a percentage of the integral’s true value. II. Using Simpson’s Rule a. Estimate the integral with n = 4 steps and find an upper bound for b. Evaluate the integral directly and find c. Use the formula ( /( true values)) × 100 to express as a percentage of the integral’s true value.

Problem 1E Estimating Integrals The instructions for the integrals in Exercises 1–10 have two parts, one for the Trapezoidal Rule and one for Simpson’s Rule. I. Using the Trapezoidal Rule a. Estimate the integral with n = 4 steps and find an upper bound for b. Evaluate the integral directly and find c. Use the formula ( /( true values)) × 100 to express as a percentage of the integral’s true value. II. Using Simpson’s Rule a. Estimate the integral with n = 4 steps and find an upper bound for b. Evaluate the integral directly and find c. Use the formula ( /( true values)) × 100 to express as a percentage of the integral’s true value.

Problem 8E Estimating Integrals The instructions for the integrals in Exercises 1–10 have two parts, one for the Trapezoidal Rule and one for Simpson’s Rule. I. Using the Trapezoidal Rule a. Estimate the integral with n = 4 steps and find an upper bound for b. Evaluate the integral directly and find c. Use the formula ( /( true values)) × 100 to express as a percentage of the integral’s true value. II. Using Simpson’s Rule a. Estimate the integral with n = 4 steps and find an upper bound for b. Evaluate the integral directly and find c. Use the formula ( /( true values)) × 100 to express as a percentage of the integral’s true value.

Problem 11E Estimating the Number of Subintervals 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 and (b) Simpson’s Rule.

Problem 10E Estimating Integrals The instructions for the integrals in Exercises 1–10 have two parts, one for the Trapezoidal Rule and one for Simpson’s Rule. I. Using the Trapezoidal Rule a. Estimate the integral with n = 4 steps and find an upper bound for b. Evaluate the integral directly and find c. Use the formula ( /( true values)) × 100 to express as a percentage of the integral’s true value. II. Using Simpson’s Rule a. Estimate the integral with n = 4 steps and find an upper bound for b. Evaluate the integral directly and find c. Use the formula ( /( true values)) × 100 to express as a percentage of the integral’s true value.

Problem 9E Estimating Integrals The instructions for the integrals in Exercises 1–10 have two parts, one for the Trapezoidal Rule and one for Simpson’s Rule. I. Using the Trapezoidal Rule a. Estimate the integral with n = 4 steps and find an upper bound for b. Evaluate the integral directly and find c. Use the formula ( /( true values)) × 100 to express as a percentage of the integral’s true value. II. Using Simpson’s Rule a. Estimate the integral with n = 4 steps and find an upper bound for b. Evaluate the integral directly and find c. Use the formula ( /( true values)) × 100 to express as a percentage of the integral’s true value.

Problem 12E Estimating the Number of Subintervals 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 and (b) Simpson’s Rule.

Problem 13E Estimating the Number of Subintervals 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 and (b) Simpson’s Rule.

Problem 15E Estimating the Number of Subintervals 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 and (b) Simpson’s Rule.

Problem 14E Estimating the Number of Subintervals 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 and (b) Simpson’s Rule.

Problem 16E Estimating the Number of Subintervals 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 and (b) Simpson’s Rule.

Problem 17E Estimating the Number of Subintervals 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 and (b) Simpson’s Rule.

Problem 18E Estimating the Number of Subintervals 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 and (b) Simpson’s Rule.

Problem 19E Estimating the Number of Subintervals 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 and (b) Simpson’s Rule.

Problem 20E Estimating the Number of Subintervals 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 and (b) Simpson’s Rule.

Problem 22E Estimating the Number of Subintervals 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 and (b) Simpson’s Rule.

Problem 23E Estimates with Numerical Data Volume of water in a swimming pool A rectangular swimming pool is 30 ft wide and 50 ft long. The accompanying table shows the depth h(x) of the water at 5-ft intervals from one end of the pool to the other. Estimate the volume of water in the pool using the Trapezoidal Rule with n = 10 applied to the integral

Problem 21E Estimating the Number of Subintervals 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 and (b) Simpson’s Rule.

Problem 24E Estimates with Numerical Data Distance traveled The accompanying table shows time-tospeed data for a sports car accelerating from rest to 130 mph. How far had the car traveled by the time it reached this speed? (Use trapezoids to estimate the area under the velocity curve, but be careful: The time intervals vary in length.)

Problem 25E Estimates with Numerical Data Wing design The design of a new airplane requires a gasoline tank of constant cross-sectional area in each wing. A scale drawing of a cross-section is shown here. The tank must hold 5000 lb of gasoline, which has a density of 42 lb/ft3 .Estimate the length of the tank by Simpson’s Rule.

Problem 29E Theory and Examples Prove that the sum T in the Trapezoidal Rule for is a Riemann sum for ƒ continuous on [a, b]. (Hint: Use the Intermediate Value Theorem to show the existence of ck in the subinterval satisfying

Problem 26E Estimates with Numerical Data Oil consumption on Pathfinder Island A diesel generator runs continuously, consuming oil at a gradually increasing rate until it must be temporarily shut down to have the filters replaced. Use the Trapezoidal Rule to estimate the amount of oil consumed by the generator during that week.

Problem 28E Theory and Examples The error function The error function, important in probability and in the theories of heat flow and signal transmission, must be evaluated numerically because there is no elementary expression for the antiderivative of a. Use Simpson’s Rule with n = 10 to estimate erf (1). b. In [0, 1], Give an upper bound for the magnitude of the error of the estimate in part

Problem 27E Theory and Examples Usable values of the sine-integral function The sine-integral function, “Sine integral of x” is one of the many functions in engineering whose formulas cannot be simplified. There is no elementary formula for the antiderivative of (sin t) t. The values of Si(x), however, are readily estimated by numerical integration. Although the notation does not show it explicitly, the function being integrated is the continuous extension of (sin t)/ t to the interval [0, x]. The function has derivatives of all orders at every point of its domain. Its graph is smooth, and you can expect good results from Simpson’s Rule. a. Use the fact that to give an upper bound for the error that will occur if is estimated by Simpson’s Rule with n = 4. b. Estimate by Simpson’s Rule with n = 4. c. Express the error bound you found in part (a) as a percentage of the value you found in part (b).

Problem 30E Theory and Examples Prove that the sum S in Simpson’s Rule for is a Riemann sum for ƒ continuous on [a, b]. (See Exercise 29.) Reference: Exercise 29 Prove that the sum T in the Trapezoidal Rule for is a Riemann sum for ƒ continuous on [a, b]. (Hint: Use the Intermediate Value Theorem to show the existence of ck in the subinterval satisfying

Problem 31E Theory and Examples Elliptic integrals The length of the ellipse turns out to be where is the ellipse’s eccentricity. The integral inthis formula, called an elliptic integral, is nonelementary except when e = 0 or 1. a. Use the Trapezoidal Rule with n = 10 to estimate the length of the ellipse when a = 1 and e = 1/2 b. Use the fact that the absolute value of the second derivative of is less than 1 to find an upper bound for the error in the estimate you obtained in part (a).

Problem 33E Theory and Examples Your metal fabrication company is bidding for a contract to make sheets of corrugated iron roofing like the one shown here. The cross-sections of the corrugated sheets are to conform to the curve If the roofing is to be stamped from flat sheets by a process that does not stretch the material, how wide should the original material be? To find out, use numerical integration to approximate the length of the sine curve to two decimal places.

Problem 32E Theory and Examples The length of one arch of the curve y = sin x is given by Estimate L by Simpson’s Rule with n = 8.

Problem 34E Theory and Examples Your engineering firm is bidding for the contract to construct the tunnel shown here. The tunnel is 300 ft long and 50 ft wide at the base. The cross-section is shaped like one arch of the curve Upon completion, the tunnel’s inside surface (excluding the roadway) will be treated with a waterproof sealer that costs $1.75 per square foot to apply. How much will it cost to apply the sealer? (Hint: Use numerical integration to find the length of the cosine curve.)

Problem 35E Theory and Examples Find, to two decimal places, the areas of the surfaces generated by revolving the curves about the x-axis

Problem 36E Theory and Examples Find, to two decimal places, the areas of the surfaces generated by revolving the curves about the x-axis

Problem 37E Theory and Examples Use numerical integration to estimate the value of For reference, sin-1 0.6 = 0.64350 to five decimal places.

Problem 38E Theory and Examples Use numerical integration to estimate the value of

Problem 39E Drug assimilation An average adult under age 60 years assimilates a 12-hr cold medicine into his or her system at a rate modeled by where y is measured in milligrams and t is the time in hours since the medication was taken. What amount of medicine is absorbed into a person's system over x12-hr period?