lesson
Updated 6 days ago ยท 1 view
Geometry gives us clean formulas for triangles, boxes, and circles โ but how do you find the exact area under a curve that bends and twists?
The big idea of calculus is simple: slice that complicated curved shape into vertical strips and approximate each strip with a simple rectangle.
Slicing the Interval
To approximate the area under a continuous function f(x) on the interval [a,b], we chop the interval into n equal subintervals.
The width of each rectangle, called ฮx (delta x), is the total interval length divided by the number of slices: ฮx=nbโaโ.
๐Create an interactive visual showing an interval [a, b] on an x-axis broken into n equal parts. Show a=[0] and b=[4] with a slider for n (n=2, 4, 8). Clearly label each partition mark x_0, x_1, x_2, ..., x_n and the bracket width Delta x = (b-a)/n. Style with clean modern UI, soft white cards (#ffffff), slate text (#1e2945), blue highlights (#2563eb), and responsive width scaling to 350px.
The endpoints of each slice are given by xiโ=a+iฮx, running from x0โ=a up to xnโ=b.
Each rectangle needs a height โ so where along each slice should we measure the curve?
Left, Right, and Midpoint Sums
A Riemann sum is the sum of these rectangle areas, named after German mathematician Bernhard Riemann who formulated it in 1854 to give calculus a rigorous definition of area.
Depending on where we evaluate the height f(xiโโ) inside each subinterval [xiโ1โ,xiโ], we get three standard approximations:
๐A 3-tab card visual comparing Left Riemann Sum (L_n), Right Riemann Sum (R_n), and Midpoint Riemann Sum (M_n) for the curve y = 0.5x^2 on [0, 4] with n = 4. Tab 1 shows rectangles touching the curve at the left corner. Tab 2 shows rectangles touching at the right corner. Tab 3 shows rectangles touching at the midpoint of each interval. Display formulas clearly: L_n = sum f(x_{i-1}) * delta_x, R_n = sum f(x_i) * delta_x, M_n = sum f(m_i) * delta_x. Color code heights and shaded rectangle areas.
In a Left Riemann Sum (Lnโ), we sample at the left edge xiโ1โ, giving Lnโ=โi=1nโf(xiโ1โ)ฮx.