3.6 Numerical integration  (Page 5/11)

${\displaystyle {\int }_1^2\frac{dx}{x}};$ trapezoidal rule; $n=5$

${\displaystyle {\int }_0^3\sqrt{4+x^3}dx};$ trapezoidal rule; $n=6$

${\displaystyle {\int }_0^3\sqrt{4+x^3}dx};$ Simpson’s rule; $n=3$

${\displaystyle {\int }_0^{12}{x}^{2}dx};$ midpoint rule; $n=6$

${\displaystyle {\int }_0^1{\text{sin}}^2\left(\pi x\right)dx};$ midpoint rule; $n=3$

Use the midpoint rule with eight subdivisions to estimate ${\displaystyle {\int }_2^4{x}^{2}dx}.$

Use the trapezoidal rule with four subdivisions to estimate ${\displaystyle {\int }_2^4{x}^{2}dx}.$

Find the exact value of ${\displaystyle {\int }_2^4{x}^{2}dx}.$ Find the error of approximation between the exact value and the value calculated using the trapezoidal rule with four subdivisions. Draw a graph to illustrate.

${\displaystyle {\int }_0^1{\text{sin}}^2\left(\pi x\right)dx};$ trapezoidal rule; $n=6$

${\displaystyle {\int }_0^3\frac{1}{1+x^3}dx};$ trapezoidal rule; $n=6$

${\displaystyle {\int }_0^3\frac{1}{1+x^3}dx};$ Simpson’s rule; $n=3$

${\displaystyle {\int }_0^{0.8}{e}^{\text{−}{x}^2}dx};$ trapezoidal rule; $n=4$

${\displaystyle {\int }_0^{0.8}{e}^{\text{−}{x}^2}dx};$ Simpson’s rule; $n=4$

${\displaystyle {\int }_0^{0.4}\text{sin}(x^2)dx};$ trapezoidal rule; $n=4$

${\displaystyle {\int }_0^{0.4}\text{sin}(x^2)dx};$ Simpson’s rule; $n=4$

${\displaystyle {\int }_{0.1}^{0.5}\frac{\text{cos}x}{x}dx};$ trapezoidal rule; $n=4$

${\displaystyle {\int }_{0.1}^{0.5}\frac{\text{cos}x}{x}dx};$ Simpson’s rule; $n=4$

Evaluate ${\displaystyle {\int }_0^1\frac{dx}{1+x^2}}$ exactly and show that the result is $\pi \text{/}4.$ Then, find the approximate value of the integral using the trapezoidal rule with $n=4$ subdivisions. Use the result to approximate the value of $\pi .$

Approximate ${\displaystyle {\int }_2^4\frac{1}{\text{ln}x}dx}$ using the midpoint rule with four subdivisions to four decimal places.

Approximate ${\displaystyle {\int }_2^4\frac{1}{\text{ln}x}dx}$ using the trapezoidal rule with eight subdivisions to four decimal places.

Use the trapezoidal rule with four subdivisions to estimate ${\displaystyle {\int }_0^{0.8}{x}^{3}dx}$ to four decimal places.

Use the trapezoidal rule with four subdivisions to estimate ${\displaystyle {\int }_0^{0.8}{x}^{3}dx}.$ Compare this value with the exact value and find the error estimate.

Using Simpson’s rule with four subdivisions, find ${\displaystyle {\int }_0^{\pi \text{/}2}\text{cos}(x)dx}.$

Show that the exact value of ${\displaystyle {\int }_0^{1}x{e}^{\text{−}x}dx=1-\frac{2}{e}.}$ Find the absolute error if you approximate the integral using the midpoint rule with 16 subdivisions.

Given ${\displaystyle {\int }_0^{1}x{e}^{\text{−}x}dx=1-\frac{2}{e},}$ use the trapezoidal rule with 16 subdivisions to approximate the integral and find the absolute error.