Measurements of a lake’s width were taken at 15-foot intervals, as shown:
x= 0 15 30 45 60 75 90 105 120
f(x)= 0 15 18 20 19 23 24 22 12
Estimate integral (0,120) f(x) dx with n = 4, using the left-hand approximation, the right hand approximation, and the trapezoidal approximation.
I figured out the midpoint approximation but I need some guidance with these other approximations.
To estimate the integral (0, 120) f(x) dx using the left-hand approximation, right-hand approximation, and trapezoidal approximation, we can use the information given in the table.
First, let's calculate the width of each interval. Since the measurements were taken at 15-foot intervals, the width of each interval is 15.
Now, let's calculate the height of each rectangle (left-hand approximation) and each rectangle (right-hand approximation) by using the given values of f(x).
For the left-hand approximation, we use the left endpoint of each interval as the height of the rectangle. So, the heights of the rectangles will be:
f(0) = 0, f(15) = 15, f(30) = 18, f(45) = 20, f(60) = 19, f(75) = 23, f(90) = 24, f(105) = 22, f(120) = 12
Now, we can calculate the area of each rectangle (left-hand approximation) by multiplying the width of the interval (15) with the corresponding height of the rectangle.
For example, for the first interval (0, 15), the area of the rectangle is 15 * f(0) = 0.
We repeat this calculation for each interval and sum up all the areas to estimate the integral:
Approximation using left-hand rectangles:
Integral (0, 120) f(x) dx ≈ (15 * f(0)) + (15 * f(15)) + (15 * f(30)) + (15 * f(45)) + (15 * f(60)) + (15 * f(75)) + (15 * f(90)) + (15 * f(105)) + (15 * f(120))
Similarly, for the right-hand approximation, we use the right endpoint of each interval as the height of the rectangle and calculate the areas using the same method.
Approximation using right-hand rectangles:
Integral (0, 120) f(x) dx ≈ (15 * f(15)) + (15 * f(30)) + (15 * f(45)) + (15 * f(60)) + (15 * f(75)) + (15 * f(90)) + (15 * f(105)) + (15 * f(120))
Finally, for the trapezoidal approximation, we calculate the area for each trapezoid (which is formed by connecting the adjacent rectangles) and sum up all the areas.
Approximation using trapezoids:
Integral (0, 120) f(x) dx ≈ (0.5 * 15 * (f(0) + f(15))) + (0.5 * 15 * (f(15) + f(30))) + ... + (0.5 * 15 * (f(105) + f(120)))
Calculating the above expressions will give you the estimates of the integral using each of the three approximations.