Estimate the area under the curve f(x)=x^2-4x+5 on [1,3]. Darw the graph and the midpoint rectangles using 8 partitions. Show how to calculate the estimated area by finding the sum of areas of the rectangles. Find the actual area under the curve on [1,3] using a definite integral.
The actual area is the integral of the f(x) function with x going from 1 to 3.
That is the value of x^3/3 -2x^2 +5x at x=3 minus the value at x=1.
Area = [9 - 19 + 15]-[1/3 - 2 + 5]
= 5 - 3 1/3 = 1 2/3
You will have to do the graph-drawing and numerical approximations to the integral yourself. Break the interval into eight rectangles. The width of each rectangle will be 1/4, and the height will be f(x) at the leading value of x.
To estimate the area under the curve f(x) = x^2 - 4x + 5 on [1,3] using midpoint rectangles with 8 partitions, follow these steps:
1. Draw the graph of the function f(x) = x^2 - 4x + 5 on a coordinate grid. The x-axis should range from 1 to 3, and the y-axis should cover the range of the function.
2. Divide the interval [1,3] into 8 equal subintervals. Each subinterval will have a width of Δx = (b - a) / n = (3 - 1) / 8 = 0.25.
3. Mark the midpoints of each subinterval. The midpoint of each subinterval can be found by using the formula: xᵢ = a + [(2i + 1) * Δx] / 2, where i represents the index of the subinterval starting from 0.
4. Draw rectangles for each subinterval with a width of Δx and a height equal to the function value at the corresponding midpoint. The height of each rectangle can be found by evaluating f(x) = x^2 - 4x + 5 at the x-coordinate of the midpoint.
5. Calculate the area of each rectangle by multiplying the width (Δx) by the height (f(x)) of each rectangle. This will give you the area of each rectangle.
6. Find the sum of the areas of all the rectangles. This can be done by adding up the areas of each rectangle calculated in the previous step. The sum of the areas of the rectangles will give an approximation of the area under the curve.
To find the actual area under the curve on [1,3], we can use a definite integral.
7. The definite integral of f(x) = x^2 - 4x + 5 from 1 to 3 can be written as ∫[1,3] (x^2 - 4x + 5) dx.
8. Evaluate the definite integral using the rules of integration. The antiderivative of x^2 - 4x + 5 with respect to x is (1/3)x^3 - 2x^2 + 5x.
9. Substitute the upper and lower limits into the antiderivative and calculate the difference: [(1/3)(3)^3 - 2(3)^2 + 5(3)] - [(1/3)(1)^3 - 2(1)^2 + 5(1)].
10. Simplify the expression to find the actual area under the curve on [1,3].
By following these steps, you can estimate the area under the curve using midpoint rectangles and find the actual area using a definite integral.