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This is a bit of an odd question, but I am learning how to compute lots of calc things from books so I don't have the benefit of watching how a teacher does it. I tend to be very error prone when computing things anyway and double integrals have a lot of moving parts and I'm finding it's easy to drop things, especially when I try and compute quickly (which is also why I want to avoid writing the double or triple integral many times). I am looking for the actual things people write on their papers when computing double integrals.

What I have been doing: To compute: $$ \int_0^1\int_x^{\sqrt{x}}(x+y^3)\mathrm dy\mathrm dx $$ I write:

Inner integral, variable $y$: $$ [xy+\frac{y^4}{4}]x^{\sqrt{x}}=x^{3/2}+\frac{x^2}{4}-(x^2+\frac{x^4}{4})=x^{3/2}-\frac{3x^2}{4}-\frac{x^4}{4} $$ Outer integral: $$ [\frac{2}{5}x^{5/2}-\frac{1}{4}x^3-\frac{1}{20}x^5]_0^1= \frac{2}{5}-\frac{1}{4}-\frac{1}{20}=\frac{1}{10} $$ Which I had to compute more than once (although it didn't help that the book had the wrong answer). Thanks for your help!

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Using your said integral, I will present how I would write down the solution with pen and paper (which of course is similar to yours):

$$\int_0^1\int_x^\sqrt{x} (x+y^3) dy\quad\!\!\!\!dx=\int_0^1\left(xy+\frac{y^4}{4}\right){\huge{|}}_x^\sqrt{x}dx$$ $$=\int_0^1x\sqrt{x}+\frac{x^2}{4}-\left(x^2+\frac{x^4}{4}\right)dx=\int_0^1x^{\frac{3}{2}}-\frac{3x^2}{4}-\frac{x^4}{4}dx$$ $$=\frac{2}{5}x^\frac{5}{2}-\frac{x^3}{4}-\frac{x^5}{20}{\huge{|}}_0^1=\frac{2}{5}-\frac{1}{4}-\frac{1}{20}-(0-0-0)=\frac{8}{20}-\frac{5}{20}-\frac{1}{20}=\frac{1}{10}$$

For aesthetic purposes, I would keep the integral in this form instead of first evaluating the inner integral and then the outer.

While I did not have to look over the problem for mistakes (as I had your answer to check against mine), it of course is essential to execute each step of the evaluation correctly. In other words, your answer could have been correct had you not made a silly mistake in integrating $\star$ or had you not forgotten that coefficient. This of course suggests that you would need practice solving problems on your own before consulting the solution and comparing how your thought process was similar/different than the one presented in the solution manual.

While Schaum's Outline does not explain any concepts, it may be invaluable in getting you to that level where you would reduce the frequencies of silly mistakes.

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