Consider the region bounded by y = mx, y = 0, x = 0, and x = b. (a) Find the upper and lower sums to approximate the area of the region when ∆x = b/4. (b) Find the upper and lower sums to approximate the area of the region when ∆x = b/n. (c) Find the area of the region by letting n approach infinity in both sums in part (b). Show that, in each case, you obtain the formula for the area of a triangle.
Riemann Sum
Riemann Sums is a special type of approximation of the area under a curve by dividing it into multiple simple shapes like rectangles or trapezoids and is used in integrals when finite sums are involved. Figuring out the area of a curve is complex hence this method makes it simple. Usually, we take the help of different integration methods for this purpose. This is one of the major parts of integral calculus.
Riemann Integral
Bernhard Riemann's integral was the first systematic description of the integral of a function on an interval in the branch of mathematics known as real analysis.
Consider the region bounded by y = mx, y = 0, x = 0, and x = b. (a) Find the upper and lower sums to approximate the area of the region when ∆x = b/4. (b) Find the upper and lower sums to approximate the area of the region when ∆x = b/n. (c) Find the area of the region by letting n approach infinity in both sums in part (b). Show that, in each case, you obtain the formula for the area of a triangle.
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