Area Between Two Curves
The general formula for finding the area between two curves is:
∫ T(x) - B(x) dx
I named the functions T(x) and B(x) specifically. T(x) represents the function "on top", while B(x) represents the function "on the bottom". In short, for x ∈ (a,b), T(x) ≥ B(x).
Here is an approach to use when finding areas between curves. A graphing calculator or mathematical software can be helpful in this procedure.
1. Draw the graphs. This is where a graphing calculator comes in handy.
2. Determine a and b, your limit points. Sometimes one or both of the limit points are found by finding the intersection of the two curves. Other problems will give you a third (or even fourth) condition. An example is x≥0, which makes one of our limit points x=0.
3. Determine which of the curves is T(x) and B(x). T(x) has greater numerical value than B(x) over the interval x ∈ (a, b).
4. Take the definite integral of T(x) - B(x). The result is your area.
∫ T(x) - B(x) dx = Area
About the examples and problem set: Please note that most examples in math textbooks, the numbers come out "nice and neat". Not so here, so I am using an HP Prime to demonstrate these examples and come up with answers. (any graphing calculator, HP, TI, Casio, all will do just fine!)
Find the area between:
y = -x^2 + 2
y = x^2 + 1
Here is a graph of the two curves:
Finding the limit points:
-x^2 + 2 = x^2 + 1
1 = 2*x^2
1/2 = x^2
Which implies x = -1/√2 and x = 1/√2
So our limit points are a = -1/√2 and b = 1/√2
From the graph we note that:
T(x) = -x^2 + 2
B(x) = x^2 + 1
T(x) - B(x) = -x^2 + 2 - (x^2 + 1) = -2*x^2 + 1
Taking the integral:
∫ -2*x^2 + 1 dx ≈ 0.94281
So the area for Example 1 is about 0.94281.
Find the area between the two curves:
y = 2 cos x
y = x - 2
With the condition x≥0
First we graph the two curves:
The two graphs intersect when x>0. Since we are given the additional condition that x≥0, the lower limit for this problem is a=0.
To find the upper limit, we find intersection point:
2 cos x = x - 2
2 cos x - x + 2 = 0
Using a numeric solver on a calculator, we find that the intersection point is x ≈ 1.71419. (See, the numbers in these examples aren't always as "neat").
In the region x ∈ (0, 1.71419), we see that 2 cos x > x - 2. Hence:
T(x) = 2 cos x
B(x) = x - 2
and the area is approximately
∫ 2 cos x - (x + 2) dx ≈ 3.93863
Problems to Try
Find the area between the curves of
y = x^2
y = 4
y = x^3
y = -3*x + 8
With the condition x≥0
y = 8*x^2 + 4*x - 6
y = x
Check your answers:
Areas are rounded to five decimal points:
2. 7.36017 (Hint: Upper Limit ≈ 1.51275)
3. 7.42101 (Hint: Use a calculator to find the limit points)
Any questions, please leave them in the comment box. Hope this is helpful and thanks!
This blog is property of Edward Shore. 2013