Calculus of variations and image segmentation

In calculus, the computation of the area under a continuous curve has been fundamentally simplified by applying the Fundamental Theorem of Calculus, Part 2. Rather than relying on the limiting process of summing infinitely many infinitesimal rectangles, this theorem permits direct evaluation using antiderivatives, thereby streamlining the process of definite integration.The Fundamental Theorem of Calculus, Part 2, states that if a function f(x) is continuous on a closed interval [a, b], then...

Determining the area of a region with straight edges is straightforward, as geometric formulas for rectangles, triangles, and polygons can be applied directly. However, traditional geometric methods are insufficient when a region has a curved boundary, such as the area under a function.fromThe area problem involves finding a systematic way to measure such regions. One approach to solving this problem is through approximation. Instead of attempting to compute the area exactly at the outset, the...

In many engineering and environmental applications, accumulated quantities are determined from rates that vary over time. A common example arises in water management, where a supply system pumps water into a storage tank at a rate that changes with time. Accurately determining how much water has entered the tank over a given period is essential for maintaining proper pressure, scheduling operations, and ensuring system safety.The flow rate of water into the tank is described by a time-dependent...

Definite integrals are essential tools in calculus, used to quantify accumulated change over an interval. A common physical application is calculating the total displacement from a velocity-time graph. If a velocity function, v(t), describes the motion of an object over time, the definite integral gives the net displacement between times a and b. This integral corresponds to the signed area under the velocity curve between those two points.Two fundamental properties of definite integrals aid in...

Consider two continuous functions defined on a closed interval from a to b. The region between these curves is bounded vertically by their graphs and horizontally by the endpoints of the interval. The objective is to measure the area of this region.An initial estimate of the area can be obtained by dividing the interval into a large number of narrow vertical strips of equal width. Each strip is approximated by a rectangle whose height is given by the vertical difference between the two...

In parametric calculus, a curve is described by a pair of functions, x(t) and y(t), where the parameter t often represents time. This representation enables a precise depiction of a particle's position as it moves through a plane, capturing both its trajectory and direction of motion. Analyzing the slope of the tangent line to the curve at a given point is fundamental for understanding how the particle moves.The slope of a tangent line to a parametric curve at any point is given by the...