Math Insight

Applet: Indefinite integral of interpolating polynomial

The blue curve is the graph of a polynomial $f(x)$. You can change $f$ by dragging the blue points, as $f$ is an interpolating polynomial through those points. The indefinite integral $\int f(x)dx$ of the function $f(x)$ is shown by the red curve. Since the slope of tangent line to the integral is the function itself, the integral $\int f(x)dx$ increases when the function $f$ is positive, is horizontal where the function $f$ is zero, and is decreasing where the function $f$ is negative. Since the integral $\int f(x) dx$ is determined only up to a constant, you can raise or lower the function by dragging the red point up and down. All these vertical translations of the red curve are the integral of $f$. To test your ability to estimate the integral from the function, you can uncheck the “show integral” checkbox and attempt to sketch what you think the integral is. Alternatively, you can uncheck the “show function” checkbox to test your ability to sketch the function from its integral.

Applet file: indefinite_integral_interpolating_polynomial_chebyshev_nodes.ggb

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