Section 15.5 : Triple Integrals
2. Evaluate \( \displaystyle \int_{0}^{1}{{\int_{0}^{{{z^{\,2}}}}{{\int_{0}^{3}{{y\cos \left( {{z^5}} \right)\,dx}}\,dy}}\,dz}}\)
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Start SolutionThere really isn’t all that much to this problem. All we need to do is integrate following the given order and recall that just like with double integrals we start with the “inside” integral and work our way out.
Also note that the fact that one of the limits is not a constant is not a problem. There is nothing that says that triple integrals set up as this is must only have constants as limits!
So, here is the \(x\) integration.
\[\begin{align*}\int_{0}^{1}{{\int_{0}^{{{z^{\,2}}}}{{\int_{0}^{3}{{y\cos \left( {{z^5}} \right)\,dx}}\,dy}}\,dz}} & = \int_{0}^{1}{{\int_{0}^{{{z^{\,2}}}}{{\left. {\left( {y\cos \left( {{z^5}} \right)\,x} \right)} \right|_0^3\,dy}}\,dz}}\\ & = \int_{0}^{1}{{\int_{0}^{{{z^{\,2}}}}{{3y\cos \left( {{z^5}} \right)\,\,dy}}\,dz}}\end{align*}\]Remember that triple integration is just like double integration and all the variables other than the one we are integrating with respect to are considered to be constants. So, for the \(x\) integration the \(y\)’s and \(z\)’s are all considered to be constants.
Show Step 2Next, we’ll do the \(y\) integration.
\[\begin{align*}\int_{0}^{1}{{\int_{0}^{{{z^{\,2}}}}{{\int_{0}^{3}{{y\cos \left( {{z^5}} \right)\,dx}}\,dy}}\,dz}}& = \int_{0}^{1}{{\left. {\left( {\frac{3}{2}{y^2}\cos \left( {{z^5}} \right)} \right)} \right|_0^{{z^2}}\,dz}}\\ & = \int_{0}^{1}{{\frac{3}{2}{z^4}\cos \left( {{z^5}} \right)\,dz}}\end{align*}\] Show Step 3Finally, we’ll do the \(z\) integration and note that the only way we are able to do this integration is because of the \({z^4}\)that is now in the integrand. Without that present we would not be able to do this integral.
\[\int_{0}^{1}{{\int_{0}^{{{z^{\,2}}}}{{\int_{0}^{3}{{y\cos \left( {{z^5}} \right)\,dx}}\,dy}}\,dz}} = \left. {\left( {\frac{3}{{10}}\sin \left( {{z^5}} \right)} \right)} \right|_0^1 = \require{bbox} \bbox[2pt,border:1px solid black]{{\frac{3}{{10}}\sin \left( 1 \right) = 0.2524}}\]So, not too much to do with this problem since the limits were already set up for us.