Today’s Plan

Roll Call
Start learning to evaluate more integrals!

Today’s Assignments

We’ll complete material for Learning Problems 07 today and also explore it further tomorrow in Groupwork 05.

Warm-up Activity

Setup an integral that represents the volume generated by rotating the region bounded by the curves $y=\sin x\qquad \text{and} \qquad y=0$ from $x=0$ to $x=\frac{\pi}{2}$ .

Let’s investigate the Product Rule!

Completed on Board

We will work through the solution as a class, with the instructor leading and students contributing key steps, ideas, and justifications throughout.

Integration by Parts

For two differentiable functions $u$ and $v$ (each depending on $x$ ) $\int u\;dv = uv - \int v \;du.$ For definite integrals evaluated from limits $a$ to $b$ , boundary evaluations are calculated as $\int_a^b u\;dv = uv\Big|_a^b - \int_a^b v \;du.$

Example 1

Use Integration by PartsIntegration by PartsFor two differentiable functions $u$ and $v$ (each depending on $x$ ) $\int u\;dv = uv - \int v \;du.$ For definite integrals evaluated from limits $a$ to $b$ , boundary evaluations are calculated as $\int_a^b u\;dv = uv\Big|_a^b - \int_a^b v \;du.$ to evaluate the integral. $\int x \sin(x)~dx.$

Completed on Board

We will work through the solution as a class, with the instructor leading and students contributing key steps, ideas, and justifications throughout.