Area between curves
- 2. How do we define area “between” curves? Two non-intersecting functions with arbitrary boundaries Two functions that have 2 or more points of intersection “ Triangular” regions bounded by three functions
- 3. Two non-intersecting functions with arbitrary boundaries Ex 1: Find the area bounded by f(x) = x 2 +2, g(x) = -x, x = 0, and x = 1 Sketch the region(s) Draw representative rectangle Determine dx or dy
- 4. Two functions that have 2 points of intersection Ex 2- Find the area bounded by f(x) = 2 – x 2 and g(x) = x Sketch the region Find the points of intersection f(x) = g(x) usually works Draw representative rectangle Determine dx or dy
- 5. Two functions that have more than 2 points of intersection Ex 3- Find the area between the graphs of f(x) = 3x 3 -x 2 -10x and g(x) = -x 2 + 2x Sketch the region(s) Find the points of intersection f(x) = g(x) usually works Draw representative rectangle Determine dx or dy
- 6. When do we ever use dy? Glad you asked Ex 4- Find the area bounded by x = 3 – y 2 and y = x - 1 Sketch the region Find the points of intersection Solve a system of equations Draw representative rectangle Determine dx or dy
- 7. “Triangular” regions bounded by three functions Ex 5- Find the area of the region bounded by: y = x, y = 2 – x, and y = -1 Sketch the region Find the points of intersection Solve a system of equations Can this be done geometrically? Draw representative rectangle Determine dx or dy Can this be done in terms of the other variable?
- 8. Quiz next session (45 min. maximum) Riemann Sums (left, right, midpoint, inscribed, circumscribed) Trapezoidal Rule Definite Integrals Fundamental Theorem of Calculus (I and II) Average Value of Function