Integration By Parts

Integration by parts may be thought of as deriving the area of the blue region from the total area and that of the red region. The area of the blue region is $A_1=\int_{y_1}^{y_2}x(y)dy$. Similarly, the area of the red region is $A_2=\int_{x_1}^{x_2}y(x)dx$. The total area, $A_1+A_2$_, is equal to the area of the bigger rectangle, $x_2y_2$, minus the area of the smaller one, $x_1y_1$: $\int_{y_1}^{y_2}x(y)dy+\int_{x_1}^{x_2}y(x)dx=\biggl.x_iy_i\biggl|_{i=1}^{i=2}$. Assuming the curve is smooth within a neighborhood, this generalizes to indefinite integrals $\int xdy + \int y dx = xy$, which can be rearranged to the form of the theorem: $\int xdy = xy - \int y dx$.

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Integration By Parts

Integration By Parts

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