complete the following steps for the given function, interval, and value of n. a. sketch the graph of the…

complete the following steps for the given function, interval, and value of n. a. sketch the graph of the function on the given interval. b. calculate δx and the grid points x0, x1, …, xn. c. illustrate the left and right riemann sums, and determine which riemann sum underestimates and which sum overestimates the area under the curve. d. calculate the left and right riemann sums. f(x)=2x² + 5 on 2,7; n = 5 d. the left riemann sum is . (simplify your answer.)

complete the following steps for the given function, interval, and value of n. a. sketch the graph of the function on the given interval. b. calculate δx and the grid points x0, x1, …, xn. c. illustrate the left and right riemann sums, and determine which riemann sum underestimates and which sum overestimates the area under the curve. d. calculate the left and right riemann sums. f(x)=2x² + 5 on 2,7; n = 5 d. the left riemann sum is . (simplify your answer.)

Answer

Explanation:

Step1: Calculate $\Delta x$

The formula for $\Delta x=\frac{b - a}{n}$, where $a = 2$, $b = 7$ and $n=5$. So $\Delta x=\frac{7 - 2}{5}=1$.

Step2: Determine the grid - points

The grid - points $x_i=a + i\Delta x$. So $x_0 = 2$, $x_1=2 + 1=3$, $x_2=2+2\times1 = 4$, $x_3=2 + 3\times1=5$, $x_4=2+4\times1 = 6$, $x_5=2+5\times1 = 7$.

Step3: Calculate the left - Riemann sum

The left - Riemann sum $L_n=\sum_{i = 0}^{n - 1}f(x_i)\Delta x$. We know that $f(x)=2x^{2}+5$, so: $f(x_0)=2\times2^{2}+5=2\times4 + 5=13$; $f(x_1)=2\times3^{2}+5=2\times9+5 = 23$; $f(x_2)=2\times4^{2}+5=2\times16 + 5=37$; $f(x_3)=2\times5^{2}+5=2\times25+5 = 55$; $f(x_4)=2\times6^{2}+5=2\times36+5 = 77$. $L_5=(f(x_0)+f(x_1)+f(x_2)+f(x_3)+f(x_4))\Delta x=(13 + 23+37+55+77)\times1=205$.

Answer:

$205$