Question

Using rectangles each of whose height is given by the value of the function at the midpoint of the rectangle's base (the midpoint rule following function, using first two and then four rectangles. 3/x between x = 1/9 and x = 1

Using two rectangles and four rectangles , the estimate for the area under the curve is ​

Answer 1

`\textsf{Two rectangles:\;\;\;Area} = (40)/(7)`

`\textsf{Four rectangles:\;\;\;Area} = (25)/(4)`

Explanation:

To estimate the area under the curve f(x) = 3/x on the interval [1/9, 1] using rectangles, we can use the midpoint method.

The idea is to partition the interval into subintervals, find the midpoint of each subinterval, evaluate the function at these midpoints, and then multiply the function values by the width of the subintervals to get the areas of the rectangles. Finally, sum these areas to estimate the total area under the curve.

`\hrulefill`

Using two rectangles

Divide the interval [1/9, 1] into two equal subintervals:

`\left[(1)/(9), (5)/(9)\right]\;\;\textsf{and}\;\;\left[(5)/(9), 1\right]`

Find the midpoint of each subinterval:

`\left\{(3)/(9), (7)/(9)\right\}`

Evaluate the function at these midpoints:

`f\left((3)/(9)\right)=(3)/((3)/(9))=9`

`f\left((7)/(9)\right)=(3)/((7)/(9))=(27)/(7)`

Calculate the area under the curve by multiplying the width of the rectangles (4/9) by the sum of the evaluations of the functions at the midpoints:

`A=(1-(1)/(9))/(2)\left[f\left((3)/(9)\right)+f\left((7)/(9)\right)\right]`

`A=(4)/(9)\left[9+(27)/(7)\right]`

`A=(40)/(7)`

Therefore, the area under the curve using two rectangles and the midpoint rule is 40/7 square units.

`\hrulefill`

Using four rectangles

Divide the interval [1/9, 1] into four equal subintervals:

`\left[(1)/(9), (3)/(9)\right],\left[(3)/(9), (5)/(9)\right], \left[(5)/(9), (7)/(9)\right],\left[(7)/(9), 1\right]`

Find the midpoint of each subinterval:

`\left\{(2)/(9), (4)/(9), (6)/(9), (8)/(9)\right\}`

Evaluate the function at these midpoints:

`f\left((2)/(9)\right)=(3)/((2)/(9))=(27)/(2)`

`f\left((4)/(9)\right)=(3)/((4)/(9))=(27)/(4)`

`f\left((6)/(9)\right)=(3)/((6)/(9))=(27)/(6)`

`f\left((8)/(9)\right)=(3)/((8)/(9))=(27)/(8)`

Calculate the area under the curve by multiplying the width of the rectangles (2/9) by the sum of the evaluations of the functions at the midpoints:

`A=(1-(1)/(9))/(4)\left[f\left((2)/(9)\right)+f\left((4)/(9)\right)+f\left((6)/(9)\right)+f\left((8)/(9)\right)\right]`

`A=(2)/(9)\left[(27)/(2)+(27)/(4)+(27)/(6)+(27)/(8)\right]`

`A=(25)/(4)`

Therefore, the area under the curve using four rectangles and the midpoint rule is 25/4 square units.