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About Properties of Definite Integrals
A definite integral is a function that assigns a value to the area under a curve on a given interval. The value of a definite integral is always a finite number, and the function can be computed using a variety of techniques. The most basic technique is to use a calculator or software to evaluate the function at a series of points on the interval and then sum the results.
Another technique is to use a method called Simpson’s rule. Simpson’s rule involves dividing the interval into a series of sub-intervals, and then computing a weighted average of the function values in each sub-interval. The weights are determined by a set of weights that are assigned to the sub-intervals. The weights are usually chosen so that the total area under the curve is exactly equal to the value of the definite integral.
There are also a number of other methods that can be used to compute definite integrals, including the method of partitions, the trapezoidal rule, and the midpoint rule.
Definite Integral Definition
The definite integral is a function that calculates the area under a curve. The curve is typically graphed on a coordinate plane, and the definite integral is represented by the symbol ∫. The area under the curve is calculated by evaluating the function at a series of points along the curve and then adding up the results.
Properties of Definite Integrals
A definite integral is a function that calculates the area between two points on a graph. The integral assigns a value to the area, which can be positive or negative, depending on the shape of the graph. The definite integral also has certain properties that make it unique.
First, the definite integral is additive. This means that the integral of a function over a certain interval is the same as the sum of the integrals of the individual functions over each sub-interval. Second, the definite integral is continuous. This means that the value of the integral at any point within the interval is the same as the value of the integral at any other point within the interval. Finally, the definite integral is differentiable. This means that the derivative of the integral is the same as the derivative of the original function.
Proofs of Definite Integrals Proofs
of Definite Integrals Theorem 1. If a function is continuous on a closed interval, then the definite integral of that function over that interval is also continuous. Proof: Let be a continuous function on the closed interval . By definition, the definite integral of over is
Since is continuous, the limit as approaches of the right-hand side exists and is equal to . Therefore, the integral is also continuous. Theorem 2. If a function is continuous on a closed interval and differentiable on the open interval within that interval, then the definite integral of that function over the closed interval is also differentiable and its derivative is the integral of the function’s derivative on the open interval. Proof: Let be a continuous function on the closed interval and differentiable on the open interval . By the mean value theorem, there exists a point within the interval such that
Since is differentiable on the open interval , the derivative exists at and is equal to . Therefore,
Since is continuous on the closed interval and differentiable on the open interval , the derivative of the integral is also continuous on the closed interval and equal to the integral of the function’s derivative on the open interval.
