<span>up vote2down voteacceptedI think your limits of integration are incorrect. If you substitute <span><span>y=4</span><span>y=4</span></span> into <span><span><span>y2</span>−<span>x2</span>=9</span><span><span>y2</span>−<span>x2</span>=9</span></span>, you find that <span><span>x=±<span>7–√</span></span><span>x=±7</span></span>. Therefore, the two curves intersect at <span><span>x=±<span>7–√</span></span><span>x=±7</span></span>. By washer method, we have:<span><span>V<span><span>=π<span>∫<span>7√</span><span>−<span>7√</span></span></span>(4<span>)2</span>−(<span><span><span>x2</span>+9</span><span>−−−−−</span>√</span><span>)2</span>dx</span><span>=2π<span>∫<span>7√</span>0</span>16−(<span>x2</span>+9)dx</span><span>=2π<span>∫<span>7√</span>0</span>7−<span>x2</span>dx</span><span>=2π<span><span>[<span>7x−<span>13</span><span>x3</span></span>]</span><span>7√</span>0</span></span><span>=2π<span>(<span><span>14<span>7–√</span></span>3</span>)</span></span><span>=<span><span>28π<span>7–√</span></span>3</span></span></span></span><span><span>V<span>=π<span>∫<span>−7</span>7</span>(4<span>)2</span>−(<span><span>x2</span>+9</span><span>)2</span>dx</span></span><span>=2π<span>∫07</span>16−(<span>x2</span>+9)dx</span><span>=2π<span>∫07</span>7−<span>x2</span>dx</span><span>=2π<span><span>[<span>7x−<span>13</span><span>x3</span></span>]</span>07</span></span><span>=2π<span>(<span><span>147</span>3</span>)</span></span><span>=<span><span>28π7</span>3</span></span></span></span>And just for fun, let's try the shell method. Here, we have no choice but to find the volume obtained by revolving just the part of the region in the first quadrant, and doubling it.<span><span>VVVVVV</span><span><span>=2×2π<span>∫43</span>y<span><span><span>y2</span>−9</span><span>−−−−−</span>√</span>dy</span><span>=4π<span>∫43</span>y<span><span><span>y2</span>−9</span><span>−−−−−</span>√</span>dy</span><span>=4π<span><span>[<span><span>13</span>(<span>y2</span>−9<span>)<span>32</span></span></span>]</span>43</span></span><span>=4π<span><span>[<span><span>13</span>(<span>y2</span>−9<span>)<span>32</span></span></span>]</span>43</span></span><span>=4π<span>[<span><span>7<span>7–√</span></span>3</span>]</span></span><span>=<span><span>28π<span>7–√</span></span>3</span></span></span></span></span>
ratio (r) is the number that, when multiplied by the previous (n-1) term, gives the nth term of the geometric sequence (). To find the common ratio, we can take any term and divide it by its preceding term (rearrange the formula to get ). If we take 24 and divide it by 16, we get the common ratio of 3/2 (, ).