(i)  Given that \(f(x) = \sqrt {16 - {x^2}} \), copy the following table of values and
supply the missing values.

(ii)  Use these five values of the function and the trapezoidal rule to find the approximate value of \(\int\limits_0^4 {\sqrt {16 - {x^2}} \,\,dx} \)

(iii)  Draw the graph of \({x^2} + {y^2} = 16\) and shade the region whose area is represented by \(\int\limits_0^4 {\sqrt {16 - {x^2}} \,\,dx} \)

(iv)  Use your answer to part (iii) to explain why the exact value of the integral is \(4\pi \)

(v)  Use your answers in part (ii) and part (iv) to find an approximate value of \(\pi \)

The diagram shows the cross-section of a river, with depths of the river shown in metres, at 5 metre intervals. The river is 15 metres wide.

(i)  Use the trapezoidal rule to find an approximate value for the area of the cross-section.

(ii)  Water flows through this section of the river at a speed of \(0.4m{s^{ - 1}}\). Calculate the approximate volume of water that flows past this section in one hour

The diagram above shows the curve \(f(x) = 2 + \sqrt[3]{x}\). The shaded area \(A\) is bound by the curve the \(x\)-axis, \(x=-8\) and \(x = 8\).

(i)  Calculate the exact area of \(A\)

(ii)  Use two applications of the Trapezoidal Rule to find the area of \(A\).

Consider the function \(y = {3^x}\)

\[\begin{array}{ | c | c | c | c | c | c |}\hline \quad x \quad & \quad-2\quad & \quad-1\quad & \quad 0 \quad & \quad 1\quad & \quad 2 \quad \\ \hline 3^x &  & & & & \\ \hline \end{array}\] (i) Copy and complete the table above.

(ii)  Using the Trapezoidal Rule with these five function values, find an estimate for the area between \(y = {3^x}\) and the lines \(x = - 2,\,\,x = 2\) and the \(x\)-axis

The function \(f(x)\) is defined by the rule \(f(x) = 2^x\), in the domain \(-2 \leq x \leq 2\). Draw up a table of values of the function, correct to 2 decimal places, for each of the values \(x = -2,\, -1,\, 0,\, 1,\, 2\). Use these five values of the function and the trapezoidal rule to find the approximate value of \[\int_{-2}^{2} \, 2^x\, dx\]

i) Given that \(f(x) = \sqrt{16 - x^2}\), copy the following table of values and supply the missing values.

\begin{array}{ | c | c | c | c | c | c |}\hline x & 0 & 1 & 2 & 3 & 4 \\ \hline f(x) & 4.000 & \,\,?\,\, & 3.464 & \,\,?\,\, & 0.000 \\\hline\end{array}

ii) Use these five values of the function and the trapezoidal rule to find the approximate value of \(\displaystyle \int_0^4 \sqrt{16 - x^2} \, dx\)

iii) Draw the graph of \(x^2 + y^2 = 16\) and shade the region whose area is represented by \(\displaystyle \int_0^4 \, \sqrt{16 - x^2} \, dx\)

iv) Use your answer to part (iii) to explain why the exact value of the integral is \(4\pi\).

v) Use your answers in part (ii) and part (iv) to find an approximate value of \(\pi\).

Determine the approximate value of \(\int\limits_1^5 {{{\log }_{10}}x\,\,dx} \) using the trapezoidal rule and five function values. 

Use the trapezoidal rule and 5 function values to show that the approximate area bounded by \(y=e^{x^2}\), the \(x\)-axis and \(x=-2,\,x=2\) is given by \(A \approx {e^4} + 2e + 1\)

i) Given \(f(x)=\sqrt{4-x^2}\) complete this table of values, correct to 3 decimal places.

\begin{array}{|c|c|c|c|c|c|}
\hline x & 0 & 0.5 & 1 & 1.5 & 2 \\
\hline f(x) & & & & & \\
\hline
\end{array}

ii) Use the Trapezoidal rule, with four sub-intervals, to estimate the value of \(\displaystyle \int_0^2 \sqrt{4-x^2} \,d x\).

The approximate value of \(\int\limits_{ - 1}^1 {{5^x}\,dx} \) using three functions values is?

Evaluate \(\int\limits_0^2 {\sqrt {1 + {x^2}} \,dx} \) using three function values

Evaluate \(\int\limits_{ - \frac{\pi }{2}}^{\frac{\pi }{2}} {\sqrt {\cos x} \,dx} \) using five function values

Evaluate \(\int\limits_1^5 {x\,{{\log }_{10}}x\,dx} \) using five function values.

Using the trapezoidal rule the area of the adjacent figure is?

The approximate value of \(\int\limits_{ - 2}^2 {{2^x}\,dx} \) using the trapezoidal rule and five function value is?

The approximate value of \(\int\limits_0^2 {{3^{ - x}}dx} \), using the trapezoidal rule and 3 function values is?

The approximate value of \(\int\limits_2^3 {\dfrac{1}{x}} \) using the trapezoidal rule and 3 function values is?

The approximate value of \(\int\limits_4^6 {{{\log }_{10}}x\,dx} \) using the trapezoidal rule and 5 function values is ?

Using the trapezoidal rule the area of the adjacent figure is?

Using the trapezoidal rule and the below table, determine the approximate distance travelled, given that \(t\) is in seconds and \(v\) is in \(m/sec\)

\[\begin{array}{ | c | c | c | c | c | c |}\hline t & 0 & 1 & 2 & 3 & 4 \\ \hline v & 1.2& 1.6& 1.2& 2.1& 1.8\\ \hline \end{array}\]