Approximating the Area under the curve
For some reason, I believe it might be best to learn area under a curve before learning integration. When approximating the area under the curve, we use left-hand and right-hand sums, mid-point rectangles, and trapezoids. There's also some jazz about approximating area between curves, but that's another lens for a different camera.
Steps to approximating area under a curve:
Steps to approximating area under a curve:
1. Graph given function
2. Shade bounded area
3. Based on shape shaded, determine which area formula is needed (if not any of the four below, solve for area as normal)
4. If LHS/RHS, MPR, or trapezoids are needed continue.
5. Determine ∆x by calculating using ∆x formula, or looking at the graph to see what the number of subdivisions is (are the rectangles/trapezoids 1 number apart? 6? 124?)
6. Draw in your shapes.
• With RHS/LHS, with ∆x = 1, the height of the rectangles is the value of the function at the x value of it's subdivision (i.e, the height of a rectangle at [1,2] of the function 2x would be calculated by 2(1) = 2, so the height of the rectangle there would be 2. The height of a rectangle at [4,5] of the function 4x would be 4(4) = 16, so the height of the rectangle at that point would be 16.
7. Approximate the area using the appropriate formula.
8. Add units. Voila.
2. Shade bounded area
3. Based on shape shaded, determine which area formula is needed (if not any of the four below, solve for area as normal)
4. If LHS/RHS, MPR, or trapezoids are needed continue.
5. Determine ∆x by calculating using ∆x formula, or looking at the graph to see what the number of subdivisions is (are the rectangles/trapezoids 1 number apart? 6? 124?)
6. Draw in your shapes.
• With RHS/LHS, with ∆x = 1, the height of the rectangles is the value of the function at the x value of it's subdivision (i.e, the height of a rectangle at [1,2] of the function 2x would be calculated by 2(1) = 2, so the height of the rectangle there would be 2. The height of a rectangle at [4,5] of the function 4x would be 4(4) = 16, so the height of the rectangle at that point would be 16.
7. Approximate the area using the appropriate formula.
8. Add units. Voila.
Formulas
Using left/right hand rectangles
NOTE: If f(x) is increasing, the LHS will be an underestimate and the RHS will be an overestimate. If f(x) is decreasing, the LHS will be on overestimate while the RHS will be an underestimate.
Practice: Approximate the area under the curve for the equation below with both LHS and RHS.
LHS:
RHS:
using midpoint rectangles
NOTE: MPR are almost always both overestimates and underestimates.
Practice: Approximate the area under the curve for the equation below.
using trapezoids
NOTE: Trapezoids are always underestimates.
Practice: Approximate the area under the curve for the equation below.