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A substitution is a very general procedure in mathematics. Your students will have experienced it many times already in algebra before they got to calculus, e.g., when solving two equations in two unknowns. More generally, it's an example of a change of variables, in the case where there is only one variable. An example in two variables would be changing ...

9

Calculating the definite integral is just as easy as graphing. You just plot your graph on a millimeter grid paper and count the number of squares below the graph. Of course, it does not give you an exact answer, only an approximation, but so is graphing. If it is easy to discern some approximate or qualitative features of the problem, it does not mean that ...

8

Introduction I wasn't taught the partial fractions decomposition (PFD) in calculus. We didn't cover it in high school, and when I went to college, they assumed we all knew it. Somehow it was when I read the proof in van der Waerden's Modern Algebra that I understood why it was called partial fractions. I looked at it again a couple days ago, and ...

7

Integration is tricky in general because it is like trying to find the question given an answer. For differentiation, we have a set of rules that work, and if we are systematic and careful enough, we can symbolically compute any derivative using those rules. Integration is tougher because you are trying to undo those rules to arrive back at the starting ...

5

I think a good analogy would be: the equations $x = x^2 - 6$ and $x = e^x$ and $x = \cos x$ all look equally complicated; but algebraic techniques are well suited to the first one and not the other two. (The second one can be solved easily but not by algebraic techniques; the third one basically can't be solved at all symbolically.) Moreover, the equation $x ... 5 I'm wondering how to introduce the idea in a way that is intuitive or geometric. How about an introduction along the lines of adding two shapes together. Start with adding fractions to have a picture in mind: For example, one could add the fractions$\frac{1}{5}$and$\frac{1}{7}$by first drawing a rectangle with width$\frac{1}{5}$and height$1 = \frac{...

4

There are sometimes reasons to avoid $u$-substitution if you can see the antiderivative without it: in a definite integral, you don't have to reparameterize the interval's endpoints to $u$-values/units if you don't use $u$-substitution. Of course, this can also be a downside in some circumstances. The fundamental thing is, though, that $u$-substitution is ...

4

The term in parentheses in the middle expression should be the change in $x\frac{1}{x}$ from $x = 1$ to $x = 2$ (which evaluates to zero), not just $x\frac{1}{x}$ (which evaluates to $1$). But as @SueVanHattum says, we are here to defend mathematics education, not to practise it.

4

It may help to connect this with the general problem of finding formulas for datasets/graphs. After all, numerically computing the function value of the antiderivative is easy, it's finding a formula to connect those values that's hard. We have lots of real world problems like this: if you want to figure out the trajectory of a thrown object, you can start ...

4

Differentiation is a science; integration is an art. When you integrate, you have to guess what function differentiated to give the one you are looking at. Substitution is one of the methods available for re-writing the function in the hope of finding something more recognisable. Students who skip the substitution are actually showing more understanding than ...

3

This is really more of an intrinsic math question than an ed question. And it comes down to "going backwards is hard". There are just not always analytical solutions (or easy ones) to definite (or indefinite) integrals. But the same thing applies with diffyQs. Or quintic and higher polynomials. Or many algebraic expressions with a mix of logs, ...

3

If I were a teacher, this would be my response to the question: Why are some integrals easy to solve while others are tricky, complicated or downright impossible? That is a very good question, and one I've asked many times and never gotten a satisfying answer for. The best I've seen is that differentiation is almost always possible because the information ...

3

I usually reference how we are filling the volume with nested cans (they can all visualize metal food cans with the top and bottom removed) and that works pretty well. Usually when I mention this, a student will ask if cylindrical shells are like Russian nesting dolls - which I think is a good analogy as well, but not all students may know of these dolls.

2

How about telescopic collapsible camping cups?        Image from wish.com. Similar to @BrendanSullivan's suggestion.

2

This is a classic example of transforming an approximation problem from the real line to the polynomial one. Let’s image we found a really good approximation, $p(x)\approx\frac{4}{x^2+1}$. $$\int^1_0 p(x)-\frac{4}{x^2+1} = p-\pi$$ Means that $p$ is a really good approximation of $\pi$. The natural conclusion is to select a few points from $\frac{4}{x^2+1}$ ...

1

Through the fundamental theorem of calculus, the difficulty with definite integrals is often connected to the difficulty of finding indefinite integrals. You might point out that given a (suitable) set of functions, integration tends to require new functions: If you restrict yourself to polynomials of degree $\le n$, integration requires you to increase the ...

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