I agree, as discussed in a few of the other answers, that most of the time engineers do not directly use calculus (or other advanced math) very often in order to do their day to day job, while at the same time having an understanding of it is vital for a good engineer. So I won't cover the same ground again. I would add, though, that understanding advanced mathematics well enough to use it effectively can be extremely helpful in this present era in which advanced mathematical tools are now available.
For example, a program such as Mathcad allows the user to perform direct integration of a domain, and an engineer who understands how to use this properly can create some of his own extremely effective, accurate, and fast tools to solve routine problems.
As a geotechnical engineer, one problem I may often find myself solving in which this ability turns out to be most useful is the primary settlement $S_p$ of a soil layer. The settlement equation is simple:
$S_p=H_{layer}\varepsilon_v=H_{layer}\frac{\Delta e}{1+e_0}$ where $\varepsilon_v$ is vertical strain and $e$ is the void ratio of the soil.
However, it turns out that $\Delta e$ is a stress-dependent quantity, and stress varies with depth (i.e., it is a function of depth, $z$):
$\Delta e=C_c\log{\frac{\sigma^{\prime}_0+\Delta \sigma^{\prime}}{\sigma^{\prime}_0}}$, where $C_c$ is the compression index (constant), and $\sigma^{\prime}$ is effective stress.
Since $\sigma^{\prime}$ changes continuously with depth, the the usual way to do this problem is to just split the soil profile into 1 foot layers, and to use the effective stress at the center of each layer to find $S_p$ for that layer. Then you just add them up.
However, a much better, and easier, way to do this is to simply directly integrate using a tool like Mathcad! Instead of dividing a 15 foot soil column into 1 foot increments, and performing the same set of calculations at each of the 15 layers, all I have to do (a single time) is this:
- Define pore water pressure as a function of depth, $z$: $u(z)=0$ (simplest case)
- Define total stress as a function of depth, $z$: $\sigma_0(z)=\gamma_{soil}z$
- Define effective stress as a function of depth, $z$: $\sigma^{\prime}_0(z)=\sigma_0(z)-u(z)$
- Define effective stress increase as a function of depth, $z$: $\Delta\sigma^{\prime}(z)=1000\text{psf}$ (simplest case is a constant increase)
- Define change in void ratio as a function of depth, $z$: $\Delta e(z)=C_c\log{\frac{\sigma^{\prime}_0(z)+\Delta \sigma^{\prime}(z)}{\sigma^{\prime}_0(z)}}$
And finally, find the total consolidation of the layer down to any depth $z=H_{layer}$ by directly integrating the primary settlement equation:
$S_p=\int_{0}^{H_{layer}}{\frac{\Delta e(z)}{1+e_0}\text{d}z}$
This approach is quicker, more accurate, and easier than the method taught in your soil mechanics or foundations textbook. However, it requires an ability to understand and apply basic calculus in order to implement it properly.
There are loads of other examples (e.g., structural analysis of a beam in bending, groundwater flow, volumetric flow analysis of a watershed hydrograph, etc etc) in which direct integration would be a superior approach to that commonly used if the right tool is available.