Posted on September 16, 2021October 24, 2022 by 202109161108 Exercises 3.3.A (Q2) For Exercises 1-8, evaluate the given triple integral. Solution.
![\begin{aligned} &\qquad \int_{0}^{1}\int_{0}^{x}\int_{0}^{y}xyz\,\mathrm{d}z\,\mathrm{d}y\,\mathrm{d}x \\ & = \int_{0}^{1}\int_{0}^{x} \bigg( \frac{xyz^2}{2}\bigg|_{z=0}^{z=y} \bigg)\,\mathrm{d}y\,\mathrm{d}x \\ & = \int_{0}^{1}\int_{0}^{x} \frac{xy^3}{2}\,\mathrm{d}y\,\mathrm{d}x \\ & = \int_{0}^{1} \bigg( \frac{xy^4}{8}\bigg|_{y=0}^{y=x} \bigg) \,\mathrm{d}x \\ & = \int_{0}^{1} \frac{x^5}{8}\,\mathrm{d}x \\ & = \bigg[ \frac{x^6}{48} \bigg]_{0}^{1} \\ & = \frac{1}{48} \\ \end{aligned}](https://physicspupil.com/wp-content/ql-cache/quicklatex.com-494694de5af1769156ad6675597d4f03_l3.svg "Rendered by QuickLaTeX.com")
