Area

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Area

Area of a Rectangle and Using Rectangles to Approximate Area Under a Curve

Recall that the area of a rectangle is the height times the base. What if we wanted to paint a wall that has a ceiling the shape of y = x² , a flat floor and a right wall at x = 2 yards and a left wall at x = 5 yards.

$Graph of y = x^2 and the left rectangles that are under the graph between x=2 and x=5$

We can approximate the area by cutting out 6 rectangles. Since the base of the wall is 5 - 2 yards long, and there are 6 rectangles, the base of each rectangle is (5 - 2)/6 = .5 yards. The height of each rectangle is the y-coordinate of the left side of each rectangle. The x- coordinates are

2 + 0(.5), 2 + 1(.5), 2 + 2(.5), 2 + 3(.5), 2 + 4(.5), 2 + 5(.5)

so that the y coordinates are

(2 + 0(.5))²,(2 + 1(.5))², (2 + 2(.5))², (2 + 3(.5))², (2 + 4(.5))², (2 + 5(.5))²

We see that the i^threctangle has y coordinate:

height = (2 + i(.5))² = 4 + 2i + .25i²

To get the area of the i^th rectangle we multiply the height by the base:

(4 + 2i + .25i²)(.5)

Finally to get the total area we add the terms up:

S[(4 + 2i + .25i²)(.5)]

This will be a lower bound for the area.

Exercise:

Find an upper bound for the area.

Left and Right Sums

If we take the limit as i approaches infinity, We arrive at the formulae:

Left Sum

$ \lim\limits_{n \to \infty} \sum_{i=0}^{n} f(a + i \Delta x) \Delta x $

Right Sum

$ \lim\limits_{n \to \infty} \sum_{i = 1}^{n} f(a + i \Delta x) \Delta x $

Note: f(x) can be negative

Usually to compute a definite integral, we use left or right sums.

Example

Use the right sum to find

$ \int_1^3 (2x + 1 dx) $

Solution:

The right sum is

$ \lim\limits_{n \to \infty} \sum_{i=0}^{n} [ 2(1+\frac{2i}{n}) + 1] \frac{2}{n} = \lim\limits_{n \to \infty} \sum_{i=0}^{n} [ 3+\frac{4i}{n})] \frac{2}{n} $

$ = \lim\limits_{n \to \infty} \sum_{i=0}^{n} [ \frac{6}{n}] + \frac{8}{n^2}\sum_{i=0}^{n} i = \lim\limits_{n \to \infty} (6 + \frac{8}{n^2}\frac{n(n+1)}{2}) $

$ = \lim\limits_{n \to \infty} (6 + \frac{8n^2 + 8n}{2n^2}) = 6 + 4 = 10$

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\( \lim\limits_{n \to \infty} \sum_{i=0}^{n} f(a + i \Delta x) \Delta x \)

\( \lim\limits_{n \to \infty} \sum_{i = 1}^{n} f(a + i \Delta x) \Delta x \)