1. 영역 R: 0 \le y \le 1이고 y \le x \le 1. (삼각형: x=1, y=0, y=x)

2. 순서 변경: 0 \le x \le 1이고 0 \le y \le x.

\displaystyle \int_{0}^{1} \int_{0}^{x} \sin(x^2)\,dy\,dx = \displaystyle \int_{0}^{1} \left[ \sin(x^2) y \right]_{y=0}^{y=x} dx = \displaystyle \int_{0}^{1} x\sin(x^2)\,dx

3. 치환 적분: u=x^2, du=2xdx.

\displaystyle \int_{0}^{1} \sin u\dfrac{1}{2}du = \displaystyle \dfrac{1}{2} \left[ -\cos u \right]_{0}^{1} = \dfrac{1}{2} \left( -\cos 1 - (-\cos 0) \right) = \displaystyle \dfrac{1}{2}(1-\cos 1)