수치적분

Approximate the integral to three decimal places using the indicated rule.

Using Simpson’s rule with four subdivisions, find ${\int }_{0}^{\pi /2}\cos (x)dx.$

Given ${\int }_{0}^{1}x{e}^{-x}dx=1-\frac{2}{e},$ use the trapezoidal rule with 16 subdivisions to approximate the integral and find the absolute error.

Find an upper bound for the error in estimating ${\int }_{0}^{3}(5x+4)dx$ using the trapezoidal rule with six steps.

Find an upper bound for the error in estimating ${\int }_{4}^{5}\frac{1}{{(x-1)}^{2}}dx$ using the trapezoidal rule with seven subdivisions.

Find an upper bound for the error in estimating ${\int }_{0}^{3}(6{x}^{2}-1)dx$ using Simpson’s rule with $n=10$ steps.

Find an upper bound for the error in estimating ${\int }_{2}^{5}\frac{1}{x-1}dx$ using Simpson’s rule with $n=10$ steps.

Find an upper bound for the error in estimating ${\int }_{0}^{\pi }2x\cos (x)dx$ using Simpson’s rule with four steps.

Estimate the minimum number of subintervals needed to approximate the integral ${\int }_{1}^{4}(5{x}^{2}+8)dx$ with an error magnitude of less than 0.0001 using the trapezoidal rule.

Determine a value of n such that the trapezoidal rule will approximate ${\int }_{0}^{1}\sqrt{1+{x}^{2}}dx$ with an error of no more than 0.01.

Estimate the minimum number of subintervals needed to approximate the integral ${\int }_{2}^{3}(2{x}^{3}+4x)dx$ with an error of magnitude less than 0.0001 using the trapezoidal rule.

Estimate the minimum number of subintervals needed to approximate the integral ${\int }_{3}^{4}\frac{1}{{(x-1)}^{2}}dx$ with an error magnitude of less than 0.0001 using the trapezoidal rule.

Use Simpson’s rule with four subdivisions to approximate the area under the probability density function $y=\frac{1}{\sqrt{2\pi }}{e}^{-{x}^{2}/2}$ from $x=0$ to $x=0.4.$

The length of one arch of the curve $y=3\sin (2x)$ is given by $L={\int }_{0}^{\pi /2}\sqrt{1+36{\cos }^{2}(2x)}dx.$ Estimate L using the trapezoidal rule with $n=6.$

The length of the ellipse $x=a\cos (t),y=b\sin (t),0\le t\le 2\pi$ is given by $L=4a{\int }_{0}^{\pi /2}\sqrt{1-{e}^{2}{\cos }^{2}(t)}dt,$ where e is the eccentricity of the ellipse. Use Simpson’s rule with $n=6$ subdivisions to estimate the length of the ellipse when $a=2$ and $e=1/3.$

[T] Use a calculator to approximate ${\int }_{0}^{1}\sin (\pi x)dx$ using the midpoint rule with 25 subdivisions. Compute the relative error of approximation.

[T] Given ${\int }_{1}^{5}(3{x}^{2}-2x)dx=100,$ approximate the value of this integral using the trapezoidal rule with 16 subdivisions and determine the absolute error.

Given that we know the Fundamental Theorem of Calculus, why would we want to develop numerical methods for definite integrals?

Choose the correct answer. When Simpson’s rule is used to approximate the definite integral, it is necessary that the number of partitions be____

a. an even number b. odd number c. either an even or an odd number d. a multiple of 4

The “Simpson” sum is based on the area under a ____.

The error formula for Simpson’s rule depends on___.

a. $f(x)$ b. ${f}^{'}(x)$ c. ${f}^{(4)}(x)$ d. the number of steps