Estimating+Area+using+Rectangles+(left,+right,+midpoint)+and+Trapezoids

Left Riemann sum uses rectangles with the length as the left side of the rectangle to estimate the area under a curve. For a function F(x), the length can be found by using f(H) where H is the x value for the left side of the rectangle. The base(B) of the rectangle is the length of the interval defined. The area can be found using[length x width] or [f(H) x B]. The summation of the areas of each rectangle will give an estimate of the area under a curve.



Right Riemann sum uses rectangles with the length as the right side of the rectangle to estimate the area under a curve. For a function F(x), the length can be found by using f(H) where H is the x value for the right side of the rectangle. The base(B) of the rectangle is the length of the interval defined. The area can be found using[length x width] or [f(H) x B]. The summation of the areas of each rectangle will give an estimate of the area under a curve.

Mid -point Riemann sum uses rectangles with the length as the middle of the rectangle to estimate the area under a curve. For a function F(x), the length can be found by using f(H) where H is the x value for the mid-point of the rectangle. The base(B) of the rectangle is the length of the interval defined. The area can be found using[length x width] or [f(H) x B]. The summation of the areas of each rectangle will give an estimate of the area under a curve.

Trapezoidal Riemann sum uses trapezoids to estimate the area under the curve. The equation for the area of a trapezoid is 1/2(B+b)(h). The height of the trapezoid is the distance of the different intervals (change in the x-value) and the base is the f(x) value. These can then be put into the equation such that 1/2(x)[f(x)1+f(x)2]. If all the intervals are of the same length the equation 1/2(x)[f(x)1+2f(x)2+...+2f(x)n-1+f(x)n] can be used where n is the number of bases.