摘要：积分法策略

Integration can sound like a scary concept at first so I like to think of it just as the process of taking an antiderivative (indefinite integral). As we have already learned, finding an antiderivative is the reverse process of finding the derivative of a function. Lets break things into pieces. To find the derivative of a simple function we must multiply the variable's coefficient by that same variable's exponent. Then we must subtract 1 from the original exponent. This is the Power Rule. Now to find the antiderivative (indefinite integral) of a function, we will complete that process backwards. First we will add 1 to the exponent then divide the function by this new exponent.

Lets try a problem...

The next step to integration is finding definite integrals. Initially, this is the same process as finding an indefinite integral. First we add 1 to the exponent of the variable then divide the function by that new exponent.

First lets find the indefinite integral of the function.

Now to find the definite integral of $∫(2,4) 32x^7dx$ we will first take the antiderivative which we have already done and plug the number at the top of the integral sign into that function to acquire ƒ(4). Next, we plug in the number at the bottom of the integral sign to obtain ƒ(2). Finally, we will subtract ƒ(4) - ƒ(2) which will give us the definite integral of $\ ∫(2,4) 32x^7dx$.

Lets practice.

A good strategy for integrating besides just using the anti-derivative method is to use Riemann Sums. Riemann Sums divide each piece of an integral into a cross-section (a "slice" of the curve) and add up all the "slices" to get an approximation. This number is not exact unless you divide the curve into an infinite number of cross-sections, eventually getting the exact same answer as the anti-derivative method.
Let's say we have the function $x$ and we want to divide it up into 4 cross-sections from $x=0$ to $x=4$ (from $0$ to $1$, $1$ to $2$, $2$ to $3$, and $3$ to $4$). We also want to use the right endpoints as our sample points (the right y-value of each "slice"). The area of a rectangle is $length x width$, so we do $(1 x 1)+(1 x 2)+(1 x 3)+(1 x 4)=10$. The estimated area under the curve is 10 if we do right sample points.

Taking this to the next level, if we have the function f(x,y)=xy and we want to find the volume under the curve from $x=0$ to $x=3$ and $y=0$ to $y=3$, we can take the volume of 9 square prisms and add them together to get an estimate. Let's use the upper right corner as our sample points: $(1,1)$, $(1,2)$,$(1,3)$,$(2,1)$,$(2,2)$,$(2,3)$,$(3,1)$,$(3,2)$, and $(3,3)$. Plugging them into our function, we get $1+2+3+2+4+6+3+6+9=36$. The estimated volume is $36$ cubic units.