AP Calculus BC · Unit 6 Integration & Accumulation
Learning Objectives
Product Rule Reversal: Integration by Parts reverses the Product Rule $d(uv) = u dv + v du$. Rearranging gives $\int u dv = uv - \int v du$.
Visualizing u and v: We can graph a parametric curve in the $(u, v)$ plane. The bounding box $uv$ represents the total rectangular area.
Area Exchange: Instead of calculating the hard area $\int u dv$ (under the curve with respect to v), we find the easy bounding box area $uv$ and subtract the adjacent area $\int v du$ (with respect to u).
Tags
IntegrationBy PartsLIATEGeometry
ABSTRACT PARAMETRIC PLANE Bot Area = $\int v du$ Top Area = $\int u dv$
Animate Transformation
Geometric Bounds
Upper Bound (t) = 1.00
The Formal Shift
$\int u dv = uv - \int v du$
PARAMETRIC EXAMPLE $(u=t^2, v=\sqrt{t})$
Total Box Area ($uv$)1.000
Bot Area ($\int v du$)0.400
Top Area ($\int u dv$)0.600
Geometrical Proof: Look at the graph. The entire big rectangle has an area of width $u$ and height $v$. The total area is exactly $uv$. It is split cleanly into two pieces by the curve: the area integrated along the u-axis ($\int v du$) and the area integrated along the v-axis ($\int u dv$)!