Applications of Integration Area Approximations Applications of Integration

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.

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.

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 0 1 1+ 𝑥 2 𝑑𝑥 . Use 6 subintervals.

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?