Question

question 5
your mathematical work must accurately support your final answer. correct answers receive no credit without mathematically correct supporting work.
(13 points) use riemann sums to estimate the area under the graph of ( y = x^{2} ) on ( 0,6 ). use left endpoints with three rectangles ( (n = 3) ).
(3 points) correct graph of function on the xy - plane with x - axis correctly labeled.
(2 points) computation of ( delta x ).
(4 points) numberline with beginning point, ending point, and all gridpoints.
(3 points) computation of area for the three rectangles.
(1 point) the estimation.
click to proceed when you have completed the work on your own paper.

Explanation:

Step1: Compute \(\Delta x\)

The formula for \(\Delta x=\frac{b - a}{n}\), where \(a = 0\), \(b=6\) and \(n = 3\).
\(\Delta x=\frac{6-0}{3}=2\)

Step2: Find the left - endpoints

The sub - intervals are \([0,2]\), \([2,4]\), \([4,6]\). The left - endpoints are \(x_0=0\), \(x_1 = 2\), \(x_2=4\)

Step3: Compute the function values at left - endpoints

For \(y=x^{2}\), when \(x = 0\), \(y_0=0^{2}=0\); when \(x = 2\), \(y_1=2^{2}=4\); when \(x = 4\), \(y_2=4^{2}=16\)

Step4: Compute the Riemann sum

The Riemann sum \(S=\sum_{i = 0}^{2}f(x_i)\Delta x\)
\(S=(0 + 4+16)\times2\)
\(S=(20)\times2\)

Answer:

\(40\)