Questions: Let f be the function given by f(x)=9 x. If four subintervals of equal length are used, what is the right Riemann sum approximation for the integral from 0 to 2 of f(x) dx? a. 22.5 b. 45 c. 54 d. 90

Let f be the function given by f(x)=9 x. If four subintervals of equal length are used, what is the right Riemann sum approximation for the integral from 0 to 2 of f(x) dx?
a. 22.5
b. 45
c. 54
d. 90

Transcript text: Let $f$ be the function given by $f(x)=9 x$. If four subintervals of equal length are used, what is the right Riemann sum approximation for $\int_{0}^{2} f(x) d x$ ? a. 22.5 b. 45 c. 54 d. 90

Solution

Solution Steps

Step 1: Determine the length of each subinterval

The interval $[0, 2]$ is divided into four subintervals of equal length. The length of each subinterval is: \[ \Delta x = \frac{2 - 0}{4} = 0.5 \]

Step 2: Identify the right endpoints of the subintervals

The right endpoints of the subintervals are: \[ x_1 = 0.5, \quad x_2 = 1.0, \quad x_3 = 1.5, \quad x_4 = 2.0 \]

Step 3: Calculate the function values at the right endpoints

Evaluate the function $f(x) = 9x$ at each right endpoint: \[ f(x_1) = 9(0.5) = 4.5 \] \[ f(x_2) = 9(1.0) = 9.0 \] \[ f(x_3) = 9(1.5) = 13.5 \] \[ f(x_4) = 9(2.0) = 18.0 \]

Step 4: Compute the right Riemann sum

Multiply each function value by $\Delta x$ and sum them up: \[ \text{Right Riemann Sum} = f(x_1)\Delta x + f(x_2)\Delta x + f(x_3)\Delta x + f(x_4)\Delta x \] \[ \text{Right Riemann Sum} = 4.5(0.5) + 9.0(0.5) + 13.5(0.5) + 18.0(0.5) \] \[ \text{Right Riemann Sum} = 2.25 + 4.5 + 6.75 + 9.0 = 22.5 \]

The right Riemann sum approximation for $\int_{0}^{2} f(x) \, dx$ is 22.5.