The area problem

THE AREA PROBLEM

Let as first attempt to solve the area problems given a function f that is continuous and non‒negative on an intervals [a, b] find the area between the graph of f and the intervals [a, b] on the x-axis.

This means that S, illustrated in is bounded by the graph of a continuous function f[where f(x) ≥ 0], the vertical lines x = a and x = b and the x-axis

For a rectangle, the area is defined as the product of the length and the width. The area of the triangle is half the base times the height.

The area of the polygon is found by dividing it into triangles adding the areas of the triangles.

However it is not so easy to find the area of a region with curved sides. But part of the area problem is to make this intuitive idea precise by giving an exact definition of area.

Recall that defining a tangent we first approximated the slope of the tangent line by slope of secant lines and then we took the limit of these approximations

We pursue a similar idea for areas. We first approximate the regions S by the rectangular and then we take the limit of the areas of these rectangular as we increases the number of rectangular.

The concept conceived from the above example may be applied to the more general region S shown in the figure 4.1 subdividing the region S into n scripts S₁, S₂, ..., S_n of equal width as in figure 4. 2.

The width of the interval [a, b] is b ‒ a, so the width of the each of the n scripts

 ∆x = (b‒a) / n

These scripts divide the intervals [a, b] into n sub intervals.

 [x₀, x₁], [x₁, x₂], [x₂, x₃], ..., [x_n‒1,x_n], where x₀ = a and x_n=b

The right endpoints of the subintervals are x₁ = a + ∆x, x₂ = a + 2∆x + ...,x_n = a + n∆x let's approximate the i^th strip S_i by a rectangle with width ∆x and height f(x_i) which is the value of f at the right endpoints.

Then the area of the i^th rectangle is f(x_i)∆x. Therefore, the area S is approximated by the sum of the area of these rectangles and is given by

 R_n = f(x₁)∆x + f(x₂)∆x + ….. + f(x_i)∆x + ….. + f(x_n)∆x

As n→ ∞, R_n→ A (Area of S)

Definition

The area A of the region S that lies under the graph of the continuous function f is the limit of the sum of the areas of approximation rectangles:

Similarly, assuming that f is continuous, the area A of S can be obtained by considering the rectangles with left endpoints as given below

Instead of using left endpoints or right endpoints, we could take the height of the i^th rectangle to be the value of f at any number x_i* in the i^th subinterval [x_i‒1,x¡].

We call the number x₁*, x^*₂, ..., x_n* the sample points as shown in figure 4.3. So more general expressions for the area of the S is

 A = lim_n→∞ [ f(x₁*)∆x + f(x₂*)∆x + ….. + f(x_i*)∆x + …. + f(x_n*)∆x ]

Example 1. Find the approximate area L₄ and R₄ for f(x) = x² between x = 0 and x = 1.

Solution: Given that f(x) = x², a = 0, b = 1 and n = 4.

∆x = (b‒a)/n = (1‒0) / 4 = 1/4

Hence the interval is subdivided into four equal parts as

The left end points are 0,1/4,1/2,3/4.

The right end points are 1/4,1/2,3/4,1

The values of the function at left end points of the intervals are

The sum of the areas of the lower approximate rectangles is

The values of the function at right end points of the intervals are

The sum of the areas of the upper approximate rectangles is

 = 15/32 = 0.46875

Example 2. Find the approximate area for f(x) = x² between x = 0 and x = 1 using eight approximate rectangles at left endpoints and right end‒points.

Solution: Given that f(x) = x², a = 0, b = 1 and n = 8.

∆x = (b‒ a)/n = (1‒0) / 8 = 1/8

Hence the interval is subdivided into eight parts as

[0,1/8],[1/8,2/8], …, [7/8,1]

The left end points are

 0,1/8,2/8, …,7/8

The right end points are

 1/8,2/8, …,7/8,1

To find L₈:

The values of the function at the left end points of the intervals are

The sum of the areas of the lower approximate rectangles is

To find R₈:

The values of the function at the right end points of the intervals are

The sum of the areas of the upper approximate rectangles is

EXERCISE

1. Estimate the area under the curve f(x) = cos x 0 ≤ x ≤ π/2 using four approximate rectangle both left end points and right end points.

Ans: L₄=0.3767π, R₄ = 0.2516π.

2. Estimate the area under the curve f(x) = √x 0 ≤ x ≤ 4 using four approximate rectangle both left end points and right end points.

Ans: L₄ = 4.146, R₄ = 6.146.

3. Estimate the area under the curve f(x)=1+x²  ‒1≤ x ≤2 using four and six approximate rectangle both left end points and right end points.

Ans: L₄=5.15625, R₄ = 7.40625, L₆ = 5.375, R₆ = 6.875.

4. Evaluate the upper and lower sum for f(x) = 1 / (1+x²) 0≤x≤1 with n = 10.

Ans: L₁₀=0.75998, R₁₀ = 0.80998.