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Using the Right Riemann Sum to Find the Area Under a Curve

Let $f(x)=\sqrt[\phantom{2}]{4-x^2}.$

$($a$)$ Sketch a graph of $f$ and the rectangles that make up the right Riemann sum $R_4$ on the interval $0,2.$

This can be done by hand.

$($b$)$ Use the function you created in the previous problem to calculate $R_4.$

$($c$)$ The area below the curve of a continuous function $f$ on the interval $a,b$ can be found by calculating

what is called the definite integral of $f$ over $a,b.$ The definite integral ${\displaystyle \underset{a}{\overset{b}{}}}f(x)dx$ can be calculated

with the following limit

${\displaystyle \underset{a}{\overset{b}{}}}f(x)dx=$ Area Under the Curve $=\underset{n}{lim}{R}_n$

Calculate appropriate right Riemann sums to find the area under the curve $y=f(x)$ on the interval

$0,2.$ Does the value of this area have a special name?