The following table gives the approximate amount of emissions, E, of nitrogen oxides in millions of metric tons per year in the US. Let t be the number of years since 1940 and E=f(t).
1940 1950 1960 1970 1980 1990
t 0 10 20 30 40 50
E 6.8 9.2 13.4 18.1 20.8 19.2
Estimate the integral: ∫^50_0f(t)dt≈
Using the trapezoidal rule, we can estimate the integral as:
∫^50_0f(t)dt ≈ (10/2)(6.8 + 9.2) + (10/2)(9.2 + 13.4) + (10/2)(13.4 + 18.1) + (10/2)(18.1 + 20.8) + (10/2)(20.8 + 19.2)
Simplifying and computing, we get:
∫^50_0f(t)dt ≈ 940 million metric tons
Therefore, an estimate for the integral is 940 million metric tons.
To estimate the integral ∫^50_0f(t)dt, we can use the trapezoidal rule. The trapezoidal rule approximates the area under the curve by dividing it into trapezoids.
Given the table, we have the following data:
t = [0, 10, 20, 30, 40, 50]
E = [6.8, 9.2, 13.4, 18.1, 20.8, 19.2]
Using the trapezoidal rule, we can approximate the integral as follows:
∫^50_0f(t)dt ≈ Δt/2 * (E[0] + 2(E[1] + E[2] + E[3] + E[4]) + E[5])
where Δt is the difference in years between each data point, which in this case is 10 years.
∫^50_0f(t)dt ≈ 10/2 * (6.8 + 2(9.2 + 13.4 + 18.1 + 20.8) + 19.2)
∫^50_0f(t)dt ≈ 5 * (6.8 + 2(9.2 + 13.4 + 18.1 + 20.8) + 19.2)
∫^50_0f(t)dt ≈ 5 * (6.8 + 2(61.5) + 19.2)
∫^50_0f(t)dt ≈ 5 * (6.8 + 123 + 19.2)
∫^50_0f(t)dt ≈ 5 * (149 + 19.2)
∫^50_0f(t)dt ≈ 5 * (168.2)
∫^50_0f(t)dt ≈ 841
Therefore, the estimated value of the integral ∫^50_0f(t)dt is approximately 841.
To estimate the integral, we can use the trapezoidal rule. The trapezoidal rule is a numerical integration method that approximates the area under a curve by dividing it into trapezoids.
First, let's calculate the width of each trapezoid. In this case, the width would be the difference in years between consecutive data points.
For example, the width of the first trapezoid would be (10 - 0) = 10 years, since the first data point is at t = 0 and the second data point is at t = 10. Similarly, the width of the second trapezoid would be (20 - 10) = 10 years.
Next, let's calculate the average height of each trapezoid. This can be done by averaging the emissions values of the two data points that form each trapezoid.
For the first trapezoid, the average height would be (6.8 + 9.2) / 2 = 8.0 million metric tons per year. Similarly, for the second trapezoid, the average height would be (9.2 + 13.4) / 2 = 11.3 million metric tons per year.
Repeat this process for each trapezoid. The calculation for each trapezoid would look like this:
Trapezoid 1:
Width = 10 years
Average height = (6.8 + 9.2) / 2 = 8.0 million metric tons per year
Area = Width * Average height = 10 * 8.0 = 80 million metric tons
Trapezoid 2:
Width = 10 years
Average height = (9.2 + 13.4) / 2 = 11.3 million metric tons per year
Area = Width * Average height = 10 * 11.3 = 113 million metric tons
Repeat this process for all the trapezoids.
Once you have calculated the area for each trapezoid, add them all together to get the estimated integral:
∫^50_0f(t)dt ≈ 80 + 113 + ... (add the areas of all the trapezoids) = total estimated integral.
Note: Since we're using the trapezoidal rule, this is an approximation of the actual integral. The more data points you have, the more accurate the approximation will be.