How Can You Find the Area Under a Curve?
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How Can You Find the Area Under a Curve?

By Josie Miller

Let’s explore this age-old question by finding the area under the curve of the parabola

To find the exact value of the area, we can divide it into infinitely small rectangles. Assume that the curve is divided into n small rectangles, and, for any given rectangle, its height is determined by the right point on the parabola (right endpoint of the rectangle). The width of any given rectangle is , where is the “change in x” or width. To find the height, we first find the right endpoint. For any rectangle, the x-coordinate of the endpoint is .  The height of the right endpoint can be found as that is the y-coordinate of the on the curve. Simplifying,  .

The area of a rectangle is length times width, so the area of any rectangle is:  .  However, the goal is to find the area of all i rectangles until n. So, . This summation can be represented using sigma notation: . This expression can be simplified by pulling out the constant (unchanging values): . Now, the formula involves a sum of squares, which can be simplified given   to

The above expression can be simplified through clever algebraic manipulation: . This equation gives the approximate area given a finite number n. To get the exact area, treat the area under the curve as divided into infinitely small rectangles. In other words, n approaches infinity. This means that becomes infinitely small, and can be concluded as equal to 0. If you plug in 0 for n,  you’ll get: So the exact area under the curve from 0 to 3 is equal to 9! This neat technique works for any polynomial over a given interval.


References

Smith, G. (2007, December 19). Isaac Newton (Stanford Encyclopedia of Philosophy). Stanford Encyclopedia of Philosophy. https://plato.stanford.edu/entries/newton/

Ximera. (n.d.). Approximating area with rectangles – Ximera. Ximera. https://ximera.osu.edu/mooculus/calculus1/approximatingTheAreaUnderACurve/digInApproximatingAreaWithRectangles