You have an unknown function that is monotone increasing for 1<x<5 and have the following information about the function values. With the clear understanding that there is no way to get an exact integral, how would you try and approximate the area under the curve?
X=(1, 1.2, 1.6, 2.8, 2.9, 3.3, 3.6, 4, 4.2, 4.8, 5)
Y = (3.170, 3.505, 4.317, 8.516, 9.041, 11.528, 13.885, 17.873, 20.312, 29.982, 32.724)
area= sum y*deltax
start at x=1, make a table
deltax yavg
.2 (6.675/2=?)
.4 (3.505+4.317)/2
and so on, then add the products of yavg*dx
To approximate the area under the curve of the unknown function, you can use numerical methods such as the trapezoidal rule or Simpson's rule. These methods involve dividing the interval [1, 5] into smaller sub-intervals and approximating the area under the curve by calculating the areas of trapezoids or Simpson's rule.
Here's how you can use the trapezoidal rule to approximate the area:
1. Identify the number of sub-intervals you want to divide the interval [1, 5] into. The more sub-intervals you use, the more accurate the approximation will be. Let's say you want to use 5 sub-intervals.
2. Calculate the width of each sub-interval by dividing the length of the interval by the number of sub-intervals. In this case, the width would be (5-1)/5 = 1.
3. Define the function values within each sub-interval by using the given coordinates (X, Y). For example, within the first sub-interval (1 to 2), the function values would be (3.170, 3.505).
4. Apply the trapezoidal rule formula to each sub-interval to calculate the area. The formula is (width/2) * (f(a) + f(b)), where f(a) and f(b) are the function values at the endpoints of the sub-interval.
5. Add up the areas calculated from each sub-interval to approximate the total area under the curve.
Using the trapezoidal rule with 5 sub-intervals, the calculation would look like this:
Area ≈ (1/2) * (f(1) + f(1.2)) + (1/2) * (f(1.2) + f(1.6)) + (1/2) * (f(1.6) + f(2)) + (1/2) * (f(2) + f(2.8)) + (1/2) * (f(2.8) + f(2.9)) + (1/2) * (f(2.9) + f(3.3)) + (1/2) * (f(3.3) + f(3.6)) + (1/2) * (f(3.6) + f(4)) + (1/2) * (f(4) + f(4.2)) + (1/2) * (f(4.2) + f(4.8)) + (1/2) * (f(4.8) + f(5))
By substituting the corresponding function values from the given data, you can calculate the approximate area under the curve using the trapezoidal rule with 5 sub-intervals.