Question

How do you find the area `y = f(x) = 4/x^2` , from 1 to 2 using ten approximating rectangles of equal widths and right endpoints?

Answer 1

See the explanation.

Explanation:

`y = f(x) = 4/x^2` , on `[1,2]` using `n=10` approximating rectangles of equal widths and right endpoints.

`Deltax = (b-a)/n = (2-1)/10=0.1`

The bases of the rectangles are the subintervals:

`[1,1.1], [1.1,1.2], [1.2,1.3], . . . , [1.8,1.9], [1.9,2]`

The right endpoints are :

`1.1, 1.2, 1.3, . . . , 1.9, 2`

Use these to find the height of the rectangles:

`f(1.1)=4/1.1^2, f(1.2)=4/1.2^2, . . . , f(1.9) = 4/1.9^2, f(2)=4/2^2`

So the areas of the rectangles are `f(x_i)Deltax` for these various `x_i` and the Riemann sum is:

`4/1.1^2Delta x+4/1.2^2 Delta x+ * * * +4/1.9^2Delta x+4/2^2Delta x`

`= (4/1.1^2+4/1.2^2 + * * * +4/1.9^2+4/2^2)Delta x`

`= (4/1.1^2+4/1.2^2 + * * * +4/1.9^2+4/2^2)(0.1)`

Now do the arithmetic.