QQuestionMathematics
QuestionMathematics
Find the formula for the Riemann sum obtained by dividing the interval
[0,27] into n equal subintervals and using the right endpoint for each ck . Then take the limit of these sums as n→∞ to calculate the area under the curve
f(2$)= 729−x^2 over [0 ,27] .
The area under the curve over [0,27] is :
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Answer
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Step 1:: Write down the formula for a Riemann sum using the right endpoint approximation.
In this case, we have [a, b] = [0, 27] and $f(x) = 729 - x^2$.
The formula for a right endpoint Riemann sum is: for an interval [a, b].
Step 2:: Plug in the values of a, b, and $f(x)$ into the Riemann sum formula.
Plugging these into the formula from Step 1, we get:
Step 3:: Simplify the expression for $f(2$)$.
\begin{align*} \end{align*}
Step 4:: Take the limit as $n \to \infty$ to find the exact area under the curve.
\begin{align*} \text{Area} &= \lim\_{n \to \infty} S\_n \ \end{align*} To evaluate the limit, we can recognize this as the definition of the definite integral and use the fundamental theorem of calculus: \begin{align*} &= 19,683 - 2,430 \ &= 17,253 \end{align*}
Final Answer
The area under the curve $f(x) = 729 - x^1$ over $[0, 27]$ is 17,253 square units.
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