Integral of pi*(-x+2)^2 dx

1. The integral of a constant times a function is the constant times the integral of the function:

   $$\int \pi \left(2 - x\right)^{2}\, dx = \pi \int \left(2 - x\right)^{2}\, dx$$

   1. There are multiple ways to do this integral.

      Method #1

      1. Let $u = 2 - x$.

         Then let $du = - dx$ and substitute $- du$:

         $$\int \left(- u^{2}\right)\, du$$

         1. The integral of a constant times a function is the constant times the integral of the function:

            $$\int u^{2}\, du = - \int u^{2}\, du$$

            1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$:

               $$\int u^{2}\, du = \frac{u^{3}}{3}$$

            So, the result is: $- \frac{u^{3}}{3}$

         Now substitute $u$ back in:

         $$- \frac{\left(2 - x\right)^{3}}{3}$$

      Method #2

      1. Rewrite the integrand:

         $$\left(2 - x\right)^{2} = x^{2} - 4 x + 4$$

      2. Integrate term-by-term:

         1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$:

            $$\int x^{2}\, dx = \frac{x^{3}}{3}$$

         1. The integral of a constant times a function is the constant times the integral of the function:

            $$\int \left(- 4 x\right)\, dx = - 4 \int x\, dx$$

            1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$:

               $$\int x\, dx = \frac{x^{2}}{2}$$

            So, the result is: $- 2 x^{2}$

         1. The integral of a constant is the constant times the variable of integration:

            $$\int 4\, dx = 4 x$$

         The result is: $\frac{x^{3}}{3} - 2 x^{2} + 4 x$

   So, the result is: $- \frac{\pi \left(2 - x\right)^{3}}{3}$

2. Now simplify:

   $$\frac{\pi \left(x - 2\right)^{3}}{3}$$

3. Add the constant of integration:

   $$\frac{\pi \left(x - 2\right)^{3}}{3}+ \mathrm{constant}$$

The answer is:

$$\frac{\pi \left(x - 2\right)^{3}}{3}+ \mathrm{constant}$$