<p>Let&nbsp;\(f(x) = 6 + 6\sin{x}\). Part of the graph of \(f\) is shown below.</p>
<div class="graph-block" data-graph-config="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" data-width="300" data-height="200" style="width:300px;height:200px;max-width:100%;display:block;margin:0 auto;"></div>
<p>The shaded region is enclosed by the curve of \(f\), the \(x\)-axis, and the \(y\)-axis.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Solve for&nbsp;\(0 \leq x &lt; 2\pi\).<span class="inline-marks">[5]</span></div><ol class="i-list"><li><div class="list-item-paragraph">\(6 + 6\sin{x} = 6\);</div></li><li><div class="list-item-paragraph">\(6 + 6\sin{x} = 0\).</div></li></ol></li><li><div class="list-item-paragraph">Write down the exact value of the \(x\)-intercept of \(f\), for&nbsp;\(0 \leq x &lt; 2\).<span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">The area of the shaded region is \(k\). Find the value of \(k\), giving your answer in terms of \(\pi\).<span class="inline-marks">[6]</span></div></li></ol>
<p>Let&nbsp;\(g(x) = 6 + 6\sin{\left(x - \frac{\pi}{2} \right)}\). The fraph of \(f\) is transformed to the graph of \(g\).</p>
<ol class="abc-list" start="4"><li><div class="list-item-paragraph">Give a full geometric description of this transformation.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Given that&nbsp;\( \displaystyle \int_{p}^{p + \frac{3\pi}{2}} {g(x)} \mathrm{d}x = k \) and \( 0 \leq p &lt; 2\pi\), write down the two values of \(p\).<span class="inline-marks">[3]</span></div></li></ol>

Let \(f(x) = 6 + 6\sin{x}\). Part of the graph of \(f\) is shown below.

The shaded region is enclosed by the curve of \(f\), the \(x\)-axis, and the \(y\)-axis.

Write down the exact value of the \(x\)-intercept of \(f\), for \(0 \leq x < 2\).

The area of the shaded region is \(k\). Find the value of \(k\), giving your answer in terms of \(\pi\).

Let \(g(x) = 6 + 6\sin{\left(x - \frac{\pi}{2} \right)}\). The fraph of \(f\) is transformed to the graph of \(g\).

Give a full geometric description of this transformation.

Given that \( \displaystyle \int_{p}^{p + \frac{3\pi}{2}} {g(x)} \mathrm{d}x = k \) and \( 0 \leq p < 2\pi\), write down the two values of \(p\).

Functions

International Baccalaureate Standard Level Mathematics