Question

How do you estimate the area under the graph of `f(x) = sqrt x` from `x=0` to `x=4` using four approximating rectangles and right endpoints?

Answer 1

For this problem: `f(x)=sqrtx`

`a=0` and `b=4`

the number of rectangles `=n=4`

`Delta x =` the length of each subinterval = the length of each base

`Delta x = (b - a)/n = (4 - 0)/4=1`

To find all of the endpoints of subintervals, start at `a` and successively add `Delta x` until you reach `b`

All endpoints: `0, 1, 2, 3, 4`.

The right endpoints are: `1, 2, 3, 4`.

The heights at the right endpoints are:

`f(1)=sqrt1=1`
`f(2)=sqrt2`
`f(3)=sqrt3`
`f(4)=sqrt4=2`

Call the areas of the rectangles `R_1, R_2` etc: Each has area `"base" xx "height"`. Every base is `Delta x` and the heights are above, so

Then the approximation we want is:
`R_1+R_2+R_3+R_4`

`=Delta x * f(1) + Delta x * f(2) +Delta x * f(3) +Delta x * f(4)`

`=1*1+1*sqrt2+1*sqrt3+1*2=1+sqrt2+sqrt3+2=3+sqrt2+sqrt3`

`3+sqrt2+sqrt3~~3+1.414+1.732=6.146`