When does trapezoidal rule overestimate?
In this video we talk about when trapezoidal rule overestimates the area under the curve, when it underestimates the area under the curve, and when it finds exact area.
In general, when a curve is concave down, trapezoidal rule will underestimate the area, because when you connect the left and right sides of the trapezoid to the curve, and then connect those two points to form the top of the trapezoid, you’ll be left with a small space above the trapezoid. The small space is outside of the trapezoid, but still under the curve, which means that it’ll get missed in the trapezoidal rule estimate, even though it’s part of the area under the curve. Which means that trapezoidal rule will consistently underestimate the area under the curve when the curve is concave down.
The opposite is true when a curve is concave up. In that case, each trapezoid will include a small amount of area that’s above the curve. Since that area is above the curve, but inside the trapezoid, it’ll get included in the trapezoidal rule estimate, even though it shouldn’t be because it’s not part of the area under the curve. Which means that trapezoidal rule will consistently overestimate the area under the curve when the curve is concave up.
So if the trapezoidal rule underestimates area when the curve is concave down, and overestimates area when the curve is concave up, then it makes sense that trapezoidal rule would find exact area when the curve is a straight line, or when the function is a linear function.