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<p>I think you may be confusing trig substitution with basic (“u”) substitution. When you integrate an e-field or potential over a loop/line/disc/etc., you aren’t doing trig sub. Trig sub is also not part of the Calc BC curriculum. You’d need to do trig substitution when you have something like integral dx / sqrt(a^2-x^2). Your substitution then is x=a*sinθ.</p>
<p>Partial derivatives also don’t come up in the AP Physics C: E&M Course. Depending on your class, you may see them hinted at when talking about the differential relationship between e-field and potential, where E = -grad(V). The gradient operator there tells you to use partial derivatives when dealing with a multidimensional potential, but all you need for the AP exam is simply E = -dV/dx.</p>
<p>You’re right on about using trivial shapes for simplifying the line/surface integrals. As long as you can recognize the integral, it’s expected to just replace it basic geometric equations for area/volume/etc.</p>