수치적분

Use the trapezoidal rule with four subdivisions to estimate ${\int }_{0}^{0.8}{x}^{3}dx$ to four decimal places.

Use the trapezoidal rule with four subdivisions to estimate ${\int }_{0}^{0.8}{x}^{3}dx.$ Compare this value with the exact value and find the error estimate.

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

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

Show that the exact value of ${\int }_{0}^{1}x{e}^{-x}dx=1-\frac{2}{e}.$ Find the absolute error if you approximate the integral using the midpoint rule with 16 subdivisions.

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 $x=0.4.$

Use Simpson’s rule with $n=14$ to approximate (to three decimal places) the area of the region bounded by the graphs of $y=0,$ $x=0,$ and $x=\pi /2.$

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 $e=1/3.$

Estimate the area of the surface generated by revolving the curve $y=\cos (2x),0\le x\le \frac{\pi }{4}$ about the x-axis. Use the trapezoidal rule with six subdivisions.

Estimate the area of the surface generated by revolving the curve $y=2{x}^{2},$ $0\le x\le 3$ about the *x-*axis. Use Simpson’s rule with $n=6.$