Area of a curved trapezoid

For a continuous nonnegative function, the definite integral represents the exact area of the region bounded by the graph of the function, the horizontal axis, and the vertical lines at the endpoints of the interval.

Conditions: The function is continuous.; The function is nonnegative.; The interval is closed [a, b].

The sample point ξᵢ is chosen within each subinterval to determine the height of the approximating rectangle. The function value f(ξᵢ) serves as the height, while the width of the subinterval Δxᵢ serves as the base.

Conditions: The interval [a, b] is divided into subintervals.; A point ξᵢ is selected from each subinterval.

For a function that changes sign, the definite integral calculates the net signed area, subtracting the area below the x-axis from the area above it. To find the total geometric area, one must integrate the absolute value of the function, $\int$|$f(x)$|dx, ensuring all contributions are positive.

Conditions: The function $f(x)$ is continuous on [a, b].; The function $f(x)$ takes both positive and negative values in [a, b].

The area of a vertical strip is approximated by a rectangle. The height of the rectangle is determined by the function value at a chosen sample point within the strip, and the width is the width of the subinterval.

Conditions: The interval is divided into subintervals.; A sample point is chosen in each subinterval.

Increasing the number of pieces does not guarantee that the width of each piece shrinks to zero. If some pieces remain wide, the approximation error in those regions persists, preventing the Riemann sum from converging to the exact integral.

Conditions: The partition is refined by adding more pieces.; The maximum width of the partition does not necessarily tend to zero.

The Riemann sum approaches the geometric area when the function is continuous and nonnegative on the interval, and the maximum width of the partition tends to zero. These conditions ensure that the rectangles accurately fill the space under the curve without significant gaps or overlaps.

Conditions: The function f is continuous on [a, b].; The function f is nonnegative on [a, b].; The maximum width of the partition tends to zero.

The maximum width of the partition (often called the mesh) must tend to zero to ensure that the Riemann sum converges to the definite integral. Merely increasing the number of rectangles is insufficient because the widths of individual rectangles might not shrink uniformly, leaving gaps or overlaps that prevent the sum from accurately approximating the area under the curve.

Conditions: The function is continuous on the closed interval [a, b].; The partition is refined such that the maximum subinterval width tends to zero.

The notation dA=$f(x)dx$ summarizes the construction of a thin vertical strip of area. It represents the product of the function's height $f(x)$ and an infinitesimal width dx, capturing the idea that the total area is the accumulation of these infinitesimal elements.

Conditions: The function $f(x)$ is continuous and nonnegative.; The area is being approximated by vertical strips.