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

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Answer 1 · 2015-03-24T17:15:38

For this problem: $f (x) = \sqrt{x}$ $f(x)=sqrtx$

$a = 0$ $a=0$ and $b = 4$ $b=4$

the number of rectangles $= n = 4$ $=n=4$

$Delta x =$ 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:
$R_1+R_2+R_3+R_4$

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

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

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

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