Home > Calculus > Integration: the Area Problem

Why does integration find the area under a curve?

Answer 1

Let us look at the definition of a definite integral below.

Definite Integral #\int_a^b f(x) dx=lim_{n to infty}sum_{i=1}^n f(a+iDelta x)Delta x#, where #Delta x ={b-a}/n#.

If #f(x)ge0#, then the definition essentially is the limit of the sum of the areas of approximating rectangles, so, by design, the definite integral represents the area of the region under the graph of #f(x)# above the x-axis.

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Answer 2

Elijah Abram

Integration finds the area under a curve because it essentially calculates the accumulation of infinitely small segments or rectangles that approximate the shape of the curve. By summing up these infinitesimal areas, integration provides a precise measurement of the total area enclosed by the curve and the x-axis within specified boundaries. This process is based on the fundamental theorem of calculus, which links differentiation and integration, allowing for the determination of areas, among other applications.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.