How do you find the Riemann sum for #f(x) = 4 sin x#, #0 ≤ x ≤ 3pi/2#, with six terms, taking the sample points to be right endpoints?

Question

Evelyn Agard · Accepted Answer

#(pisqrt2)/2# #n=6, a=0, b=(3pi)/2# #Deltax=(b-a)/n=((3pi)/2)/6=(3pi)/12=pi/4# #S=sum_(i=1)^6 f(x_i)Deltax# #S=Deltax(f(x_1)+f(x_2)+f(x_3)+f(x_4)+f(x_5)+f(x_6))# #x_i=a+i*Deltax=0+i*Deltax=i*Deltax# #f(x_1)=f(pi/4)=4sin(pi/4)=4 sqrt2/2=2sqrt2# #f(x_2)=f(2pi/4)=4sin(pi/2)=4# #f(x_3)=f(3pi/4)=4sin(3pi/4)=4 sqrt2/2=2sqrt2# #f(x_4)=f(4pi/4)=4sin(pi)=0# #f(x_5)=f(5pi/4)=4sin(5pi/4)=4 (-sqrt2/2)=-2sqrt2# #f(x_6)=f(6pi/4)=4sin(3pi/2)=4 (-1)=-4# #S=pi/4(2sqrt2+4+2sqrt2+0-2sqrt2-4)=pi/4 2sqrt2=(pisqrt2)/2#

Paisley Johnson · Answer

undefined To find the Riemann sum for $ f(x) = 4 \sin(x) $ on the interval $ 0 \leq x \leq \frac{3\pi}{2} $ with six terms, taking the sample points to be right endpoints, follow these steps:

1. Divide the interval $ [0, \frac{3\pi}{2}] $ into six subintervals of equal width.
2. Determine the width of each subinterval: $ \Delta x = \frac{3\pi/2 - 0}{6} = \frac{\pi}{4} $.
3. Identify the right endpoints of the subintervals.
4. Evaluate $ f(x) = 4 \sin(x) $ at each right endpoint.
5. Multiply the function values by the width of the subintervals and sum them up.

The Riemann sum can be expressed as:

$$ \text{Riemann sum} = \Delta x \sum_{i=1}^{n} f(x_i) $$

Where $ \Delta x $ is the width of each subinterval, and $ f(x_i) $ is the value of the function $ f(x) $ at the right endpoint $ x_i $ of the $ i $th subinterval.

Given $ f(x) = 4 \sin(x) $, you evaluate it at the right endpoints of each subinterval:

$$ x_1 = \frac{\pi}{4}, \quad x_2 = \frac{\pi}{2}, \quad x_3 = \frac{3\pi}{4}, \quad x_4 = \pi, \quad x_5 = \frac{5\pi}{4}, \quad x_6 = \frac{3\pi}{2} $$

Then, calculate $ f(x_i) $ for each $ x_i $:

$$ f\left(\frac{\pi}{4}\right), \quad f\left(\frac{\pi}{2}\right), \quad f\left(\frac{3\pi}{4}\right), \quad f(\pi), \quad f\left(\frac{5\pi}{4}\right), \quad f\left(\frac{3\pi}{2}\right) $$

After evaluating $ f(x_i) $ for each $ x_i $, multiply each value by $ \Delta x = \frac{\pi}{4} $, and sum up the results to find the Riemann sum.