Question

question 2
estimate the area from 0 to 5 under the graph of ( f(x)=64 - x^{2} ) using 5 approximating rectangles and right endpoints.
( 265 ) units( ^{2} )
the correct answer is not listed.
( 267 ) units( ^{2} )
( 266 ) units( ^{2} )
( 263 ) units( ^{2} )

Explanation:

Step1: Calculate the width of each rectangle

The interval is from \(a = 0\) to \(b = 5\), and \(n=5\). The width \(\Delta x=\frac{b - a}{n}=\frac{5-0}{5}=1\)

Step2: Find the right - endpoints

The right - endpoints \(x_i=a + i\Delta x\), for \(i = 1,2,\cdots,5\). So \(x_1=1,x_2 = 2,x_3=3,x_4=4,x_5=5\)

Step3: Calculate the function values at the right - endpoints

\(f(x_1)=64-1^2=63\), \(f(x_2)=64 - 2^2=60\), \(f(x_3)=64-3^2=55\), \(f(x_4)=64 - 4^2=48\), \(f(x_5)=64-5^2=39\)

Step4: Calculate the sum of the areas of the rectangles

Using the formula for the right - Riemann sum \(R_n=\sum_{i = 1}^{n}f(x_i)\Delta x\). Since \(\Delta x = 1\), \(R_5=f(x_1)\times1+f(x_2)\times1+f(x_3)\times1+f(x_4)\times1+f(x_5)\times1\)
\(R_5=63 + 60+55+48+39\)
\(R_5=265\)

Answer:

265 units²