8-8 Improper Integrals Rizzi – Calc BC

Two Goals Evaluate an improper integral that has an infinite limit of integration. Evaluate an improper integral that has an infinite discontinuity. Evaluate

With your partner, do page 1

Infinite Limits of Integration 1 ∞ 𝑒 −𝑥 𝑑𝑥 Evaluate Evaluate =1

Try Example 2 with your partner. Share the workload!

Proof! Use what you know about limits to prove the following theorem

Use the properties of p-series to do Example 3 with your partner Use the properties of p-series to do Example 3 with your partner. Then use your knowledge of limits to do Example 5.

Use your calculator to evaluate the integrals in Example 12 Use your calculator to evaluate the integrals in Example 12. Using the graph, explain what is unique about these integrals.

Infinite Discontinuities – Evaluated Mathematically (Example 13) 0 1 𝑑𝑥 𝑥 3 Evaluate Evaluate

Discontinuities Inside the Interval 2 5 𝑑𝑥 𝑥−3

One More with a Discontinuity −2 2 𝑑𝑥 𝑥 2 3

Checklist Check to see if one of your limits of integration is ∞ If so, you’ll need to replace the ∞ with a variable and then take the limit after you integrate. Make sure your function is defined on both ends. If it’s not, replace the undefined one with a variable and then take the limit after you integrate. Make sure there are no discontinuities in your interval If there are, you’ll need to split the integral into two pieces.