Sue Pemberton, Julianne Hughes and, Julian Gilbey Solutions for Chapter: Integration, Exercise 6: EXERCISE 5E

Use the trapezium rule with $2$ intervals to estimate the value of the following definite integral. Give your answers correct to $2$ decimal places.

${\int }_1^7\frac{2}{\ln (x+5)}dx$.[Use $\ln 6=1.79175947,\ln 9=2.19722458,\ln 12=2.48490665$]

Use the trapezium rule with $2$ intervals to estimate the value of the following definite integral. Give your answers correct to $2$ decimal places.

${\int }_0^{\frac{1}{3}\pi }\frac{1}{2+\tan x}dx$.[Use $\tan \left(\frac{\mathrm{\pi }}{6}\right)=\frac{1}{\sqrt{3}},\tan \left(\frac{\mathrm{\pi }}{3}\right)=\sqrt{3}$ and also use $\mathrm{\pi }=3.14$]

Use the trapezium rule with $2$ intervals to estimate the value of the following definite integral. Give your answers correct to $2$ decimal places.

${\int }_2^{12}\left({\log }_{10}x\right)dx$.  [Use, ${\log }_{10}2=0.301,{\log }_{10}7=0.8450,{\log }_{10}12=1.079181246$]

The diagram shows part of the curve $y=x^2{e}^{-x}$. Use the trapezium rule with $4$ intervals to estimate the value of ${\int }_1^5{x}^2{e}^{-x}dx$, giving your answer correct to $2$ decimal places. [Use, $e=2.7$]

 Use the trapezium rule with $6$ intervals to estimate the value of ${\int }_{\frac{1}{4}\pi }^{\frac{7}{4}\pi }cosec\left(\frac{1}{2}x\right)dx$ giving your answer correct to $2$ decimal places.[Use, $\mathrm{cosec}\left(22.5°\right)=2.6131259297528$ and also use, $\mathrm{\pi }=3.14,\sqrt{2}=1.414$]

Use a sketch of the graph of $y=cosec\left(\frac{1}{2}x\right)\text{ for }0⩽x⩽2\pi$, to explain whether the trapezium rule gives an under-estimate or an over-estimate of the true value of ${\int }_{\frac{1}{4}\pi }^{\frac{7}{4}\pi }cosec\left(\frac{1}{2}x\right)dx$.[Use, $\mathrm{cosec}\left(22.5°\right)=2.6131259297528$ and also use, $\mathrm{\pi }=3.14,\sqrt{2}=1.414$]

The diagram shows the curve $y=e^x\cos x\text{ for }0⩽x⩽\frac{1}{2}\pi$. Use the trapezium rule with $3$ intervals to estimate the value of ${\int }_0^{\frac{1}{2}\pi }{e}^x\cos x dx$, giving your answer correct to $2$ decimal places. State, with a reason, whether the trapezium rule gives an under-estimate or an over-estimate of the true value of ${\int }_0^{\frac{1}{2}\pi }{e}^x\cos x dx$. [Use, $e=2.718,\mathrm{\pi }=3.14,\sqrt{3}=1.732$]

The diagram shows part of the curve $y=\frac{e^x}{2x}$. Use the trapezium rule with $4$ intervals to estimate the area of the shaded region, giving your answer correct to $2$ decimal places. State, with a reason, whether the trapezium rule gives an under-estimate or an over-estimate of the true value of the shaded area. [Use, $e=2.718$]