All Labs

Accumulation, Area & Volume Lab

Investigate definite integrals through Riemann sum approximations, compute areas between curves, and calculate volumes of revolution using disk, washer, and shell methods. Watch convergence in real time and verify the Fundamental Theorem of Calculus.

Guided Experiment: Riemann Sum Convergence

If you increase the number of rectangles in a Riemann sum, what do you predict will happen to the error between the approximation and the exact integral?

Write your hypothesis in the Lab Report panel, then click Next.

Riemann Sum Approximation

Controls

Rectangles (n)10

Results

∫02f(x) dx≈2.280000\int_{0}^{2} f(x)\,dx \approx 2.280000
Approximate
2.280000
Exact (Simpson)
2.666667
% Error
14.5000%
FTC Verification
F(2)−F(0)=2.666667F(2) - F(0) = 2.666667
n=10n = 10 rectangles, left method | Δx=0.2000\Delta x = 0.2000

Data Table

(0 rows)
#TrialFunctionMethodnApprox. ValueExact Value% Error
0 / 500
0 / 500
0 / 500

Reference Guide

Riemann Sums

A Riemann sum approximates a definite integral by dividing the area under a curve into rectangles.

∑i=1nf(xi∗)Δx≈∫abf(x) dx\sum_{i=1}^{n} f(x_i^*) \Delta x \approx \int_a^b f(x)\,dx

The sample point x* can be the left endpoint, right endpoint, or midpoint of each subinterval. As the number of rectangles increases, the approximation converges to the exact integral.

Fundamental Theorem of Calculus

The FTC connects differentiation and integration, providing a way to evaluate definite integrals exactly.

∫abf(x) dx=F(b)−F(a)\int_a^b f(x)\,dx = F(b) - F(a)

If F is an antiderivative of f, then the definite integral equals the net change in F. This lab lets you verify that Riemann sums converge to this exact value.

Area Between Curves

The area of a region bounded by two curves f(x) and g(x) on [a, b] is found by integrating the absolute difference.

A=∫ab∣f(x)−g(x)∣ dxA = \int_a^b |f(x) - g(x)|\,dx

The lab automatically finds intersection points and shades the enclosed region. The integral is computed numerically using Simpson's rule.

Volumes of Revolution

Rotating a curve around an axis produces a solid of revolution whose volume can be computed by integration.

V=π∫ab[f(x)]2 dx(disk)V = \pi \int_a^b [f(x)]^2\,dx \quad \text{(disk)}
V=2π∫abx⋅f(x) dx(shell)V = 2\pi \int_a^b x \cdot f(x)\,dx \quad \text{(shell)}

Choose disk, washer, or shell method depending on the axis of revolution. The cross-section slider shows the generating shape at any point along the interval.