I have done this question, but i can't seem to get the answer 31.
Qns:
\(\displaystyle \L \int_0^{2}\, \int_0^{\frac{6-3u}{2}}\, 3v\, +\, 2u^{2}\, + (6\, -\, 3u\, -\, 2v)^{2}\)
My Working:
I just can't seem to get the answer after expansion and i just wondering if anyone can spot the mistake.
After expanding:
\(\displaystyle \L \int_0^{2}\, \int_0^{\frac{6-3u}{2}}\, 3v\, +\, 2u^{2}\, + (6\, -\, 3u\, -\, 2v)^{2}\)
\(\displaystyle \L \int_0^{2}11u^{2}(\frac{6-3u)}{2}) + \frac{4}{3}(\frac{6-3u}{2})^{3} - \frac{21}{2} (\frac{6-3u}{2})^{2} - 36u (\frac{6-3u}{2}) + 6u (\frac{6-3u}{2}) + 36 (\frac{6-3u}{2}) \, du\)
\(\displaystyle \L =\int_0^{2} 33u^{2} - \frac{33}{2} u^{2} + \frac{1}{6}(216 - 324u + 162u^{2} + 27u^{3}) - \frac{21}{8}(36 - 36u + 9u^{2})\)
. . . . .\(\displaystyle \L - (108u - 54^{2}) + 6u(36 - 36u + 9^{2}) + (108 - 54u) \,du\)
\(\displaystyle \L =\int_0^{2} 33u^{2} - \frac{32}{2}u^{3} + 36 - 54u + 27u^{2} + \frac{27}{6}u^{3} - \frac{756}{8} + \frac{756}{8}u - \frac{189}{8}u^2\)
. . . . .\(\displaystyle \L - 108u + 54u^{2} + 216u -216u^{2} + 54^{3} + 108 - 54u\)
After much calculation, here are the last remaining steps:
\(\displaystyle \L \int_0^{2} -\frac{1005}{8}u^{2} + 42u^{3} + \frac{189}{2}u + \frac{99}{2}\)
\(\displaystyle \L =[42(\frac{u^{4}}{4}) - \frac{1005}{8}(\frac{u^{3}}{3}) + \frac{189}{2}(\frac{u^{2}}{2}) + \frac{99}{2}u]_0^{2}\)
\(\displaystyle \L =121\)
Qns:
\(\displaystyle \L \int_0^{2}\, \int_0^{\frac{6-3u}{2}}\, 3v\, +\, 2u^{2}\, + (6\, -\, 3u\, -\, 2v)^{2}\)
My Working:
I just can't seem to get the answer after expansion and i just wondering if anyone can spot the mistake.
After expanding:
\(\displaystyle \L \int_0^{2}\, \int_0^{\frac{6-3u}{2}}\, 3v\, +\, 2u^{2}\, + (6\, -\, 3u\, -\, 2v)^{2}\)
\(\displaystyle \L \int_0^{2}11u^{2}(\frac{6-3u)}{2}) + \frac{4}{3}(\frac{6-3u}{2})^{3} - \frac{21}{2} (\frac{6-3u}{2})^{2} - 36u (\frac{6-3u}{2}) + 6u (\frac{6-3u}{2}) + 36 (\frac{6-3u}{2}) \, du\)
\(\displaystyle \L =\int_0^{2} 33u^{2} - \frac{33}{2} u^{2} + \frac{1}{6}(216 - 324u + 162u^{2} + 27u^{3}) - \frac{21}{8}(36 - 36u + 9u^{2})\)
. . . . .\(\displaystyle \L - (108u - 54^{2}) + 6u(36 - 36u + 9^{2}) + (108 - 54u) \,du\)
\(\displaystyle \L =\int_0^{2} 33u^{2} - \frac{32}{2}u^{3} + 36 - 54u + 27u^{2} + \frac{27}{6}u^{3} - \frac{756}{8} + \frac{756}{8}u - \frac{189}{8}u^2\)
. . . . .\(\displaystyle \L - 108u + 54u^{2} + 216u -216u^{2} + 54^{3} + 108 - 54u\)
After much calculation, here are the last remaining steps:
\(\displaystyle \L \int_0^{2} -\frac{1005}{8}u^{2} + 42u^{3} + \frac{189}{2}u + \frac{99}{2}\)
\(\displaystyle \L =[42(\frac{u^{4}}{4}) - \frac{1005}{8}(\frac{u^{3}}{3}) + \frac{189}{2}(\frac{u^{2}}{2}) + \frac{99}{2}u]_0^{2}\)
\(\displaystyle \L =121\)
