1. Applications of Integration
Area Approximations
Applications of Integration

2. Approximations
Approximate the area under the curve 𝑦= 𝑥 2 +1 from 𝑥=0 to 𝑥=4. Use 4 subintervals (n=4)
Using a left sum
Using a right sum
Using a midpoint sum
Using the Trapezoidal Rule
Using rectangles of whatever width you choose.
Find the EXACT area.

3. Approximating Areas Approximate 0 1 1+ 𝑥 2 𝑑𝑥 . Use 6 subintervals.
Elise: Use a left sum.
Nina: Use a right sum.
Amiee: Use a midpoint sum.
Together: Use the Trapezoidal Rule.

4. Simpson’s Rule
𝑎 𝑏 𝑓 𝑥 𝑑𝑥≈ 𝑏−𝑎 3𝑛 𝑦 0 +4 𝑦 1 +2 𝑦 2 +4 𝑦 3 +2 𝑦 4 +…+4 𝑦 𝑛−1 + 𝑦 𝑛
Note: n must be EVEN to use Simpson’s Rule!
Approximate 𝑥 2 𝑑𝑥 . Use 6 subintervals.

5. Simpson’s Rule
Use Simpson’s Rule with 𝑛=2 to approximate 0 𝜋 2 cos 𝑥 𝑑𝑥.
Use Simpson’s Rule with 𝑛=4 to approximate 0 𝜋 2 cos 𝑥 𝑑𝑥.
What is the actual value of 0 𝜋 2 cos 𝑥 𝑑𝑥?