Estimate the are 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.
To estimate the area under the curve using midpoint rectangles method, follow these steps:
Step 1: Draw the graph of the function f(x) = x^2 - 4x + 5 on the interval [1, 3].
To do this, plot some points on the graph:
- When x = 1, f(1) = (1^2) - 4(1) + 5 = 2
- When x = 2, f(2) = (2^2) - 4(2) + 5 = 1
- When x = 3, f(3) = (3^2) - 4(3) + 5 = 5
Using these points, you can plot the graph of the curve. Make sure the graph is smooth without any sharp corners.
Step 2: Divide the interval [1, 3] into 8 equal partitions.
To do this, find the width of each subinterval by subtracting the starting point of the interval from the ending point and dividing it by the number of partitions.
Width of each subinterval = (3 - 1) / 8 = 0.25
So, the 8 partitions for the interval [1, 3] are: [1, 1.25, 1.5, 1.75, 2, 2.25, 2.5, 2.75, 3]
Step 3: Calculate the height of each rectangle using the midpoint value of each subinterval.
Since we have 8 subintervals, find the midpoint value of each subinterval by adding the starting point of the subinterval to the width of the subinterval divided by 2.
Midpoint values for the subintervals: [1.125, 1.375, 1.625, 1.875, 2.125, 2.375, 2.625, 2.875]
Step 4: Calculate the area of each rectangle by multiplying the width by the height.
Area of each rectangle = Width of subinterval × Height of rectangle
Let's calculate the area for each rectangle:
- Rectangle 1: Area = 0.25 × f(1.125)
- Rectangle 2: Area = 0.25 × f(1.375)
- Rectangle 3: Area = 0.25 × f(1.625)
- Rectangle 4: Area = 0.25 × f(1.875)
- Rectangle 5: Area = 0.25 × f(2.125)
- Rectangle 6: Area = 0.25 × f(2.375)
- Rectangle 7: Area = 0.25 × f(2.625)
- Rectangle 8: Area = 0.25 × f(2.875)
Step 5: Sum up the areas of all the rectangles to estimate the total area under the curve.
Estimated Area = Sum of areas of all rectangles
= Area of Rectangle 1 + Area of Rectangle 2 + ... + Area of Rectangle 8
Now, calculate the estimated area by adding up the areas of all the rectangles.
To find the actual area under the curve on [1, 3] using a definite integral, integrate the function f(x) = x^2 - 4x + 5 with limits from 1 to 3:
Actual Area = ∫[1 to 3] (x^2 - 4x + 5) dx
By evaluating this definite integral, you can find the exact value of the area under the curve.