Given that \(a\) is a non-zero constant, which of the following integrals is equal to zero?

1. \(\displaystyle \int_{-a}^a x\, \cos ^{-1}(x) d x\)
2. \(\displaystyle\int_{-a}^a x^2\, \cos ^{-1}(x) d x\)
3. \(\displaystyle\int_{-a}^a x\, \tan ^{-1}(x) d x\)
4. \(\displaystyle\int_{-a}^a x^2\, \tan ^{-1}(x) d x\)

The region, \(R\), is bounded by the curves  \(y=\sin x, y=x\)  and the line  \(x=\dfrac{\pi}{2}\)  as shown in the diagram.
 

Find the area of the region \(R\).   (3 marks)

Consider the functions  \(y=f(x)\)  and  \(y=g(x)\), and the regions shaded in the diagram below. 
 

Which of the following gives the total area of the shaded regions?

1. \(\displaystyle \int_{-4}^4 f(x)-g(x)\,d x\)
2. \(\displaystyle \left|\int_{-4}^4 f(x)-g(x)\,d x\right|\)
3. \(\displaystyle \int_{-4}^{-3} f(x)-g(x)\,d x+\int_{-3}^{-1} f(x)-g(x)\,d x+\int_{-1}^1 f(x)-g(x)\,d x+\int_1^4 f(x)-g(x)\,d x \)
4. \(\displaystyle - \int_{-4}^{-3} f(x)-g(x)\,d x+\int_{-3}^{-1} f(x)-g(x)\,d x-\int_{-1}^1 f(x)-g(x)\,d x+\int_1^4 f(x)-g(x)\,d x\)

The diagram shows the graphs of the functions \(f(x)\) and \(g(x)\).
 

It is known that

\begin{aligned} & \int_a^c f(x) d x=10 \\ & \int_a^b g(x) d x=-2 \\ & \int_b^c g(x) d x=3 .\end{aligned}

What is the area between the curves  \(y=f(x)\)  and  \(y=g(x)\)  between \(x=a\) and \(x=c\) ?

1. 5
2. 7
3. 9
4. 11

The region enclosed by  `y = 2 - |x|`  and  `y = 1 - 8/(4 + x^2)`  is shaded in the diagram.
 

Find the exact value of the area of the shaded region.  (3 marks)

1. Sketch the region bounded by the curve  `y = x^2`  and the lines  `y = 16`  and  `y = 9`.  (1 mark)
   

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2. Calculate the area of this region.  (3 marks)
   

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i.  Show that  `tan((5pi)/(12)) = sqrt3 + 2`.   (2 marks)

ii. 
       
 
Hence, find the area bounded by the graph of  `f(x) = (2)/(x^2 - 4x + 8)`  shown above, the `x`-axis and the lines  `x = 0`  and  `x = 2 sqrt3 +6`.   (4 marks)