Understanding Definite Integrals: A Guide to Calculating Accumulated Area in Mathematics

Theorem: the definite integral of a continuous function on [a,b]

The definite integral of a continuous function on the interval [a, b] is a concept in calculus that represents the accumulated area between the graph of the function and the x-axis over that interval

The definite integral of a continuous function on the interval [a, b] is a concept in calculus that represents the accumulated area between the graph of the function and the x-axis over that interval.

To understand the concept of definite integrals, let’s break it down step by step:

1. Consider a continuous function f(x) defined on the interval [a, b]. The graph of this function can have positive and negative regions, as it can go above and below the x-axis.

2. The definite integral of f(x) from a to b, denoted by ∫[a, b] f(x) dx, is the signed area between the graph of f(x) and the x-axis over the interval [a, b]. The sign indicates whether the area is positive or negative depending on whether the function is above or below the x-axis over that interval.

3. To calculate the definite integral, we can divide the interval [a, b] into small subintervals. Let’s call these subintervals [x₀, x₁], [x₁, x₂], [x₂, x₃], and so on, up to the last subinterval [xₙ₋₁, xₙ]. The endpoints of these subintervals are a = x₀ < x₁ < x₂ < ... < xₙ = b. 4. We approximate the area under the curve of the function by approximating the shape of the subintervals as rectangles. The area of each rectangle is given by the height of the function multiplied by the width of the subinterval. If we represent the width of each subinterval as Δx and the height of the function at a particular point in the subinterval as f(xᵢ), then the area of each rectangle is Δx * f(xᵢ). 5. To get a more accurate approximation, we can make the subintervals narrower by taking a limit as the number of subintervals, n, approaches infinity (or in other words, taking the width of each subinterval, Δx, to approach 0). 6. Using mathematical notation, the definite integral of f(x) from a to b is defined as the limit of the sum of the areas of these rectangles as n approaches infinity: ∫[a, b] f(x) dx = limit as n approaches infinity of the sum from i = 0 to n-1 of (f(xᵢ) * Δx) 7. This limit represents the exact area under the curve of the function f(x) over the interval [a, b]. It takes into account both positive and negative areas, as the rectangles can have positive or negative widths depending on whether the function is above or below the x-axis. 8. To evaluate definite integrals, we can use various integration techniques such as the Fundamental Theorem of Calculus, integration by substitution, integration by parts, or tables of integral formulas, depending on the complexity of the function. In summary, the definite integral of a continuous function on the interval [a, b] represents the accumulated area under the curve of that function over that interval. It is a fundamental concept in calculus and has applications in various fields such as physics, engineering, economics, and more.

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