Question

Find the 4th left, right, and midpoint Riemann sums of the following functions with respect to particular partitioning of the given intervals. (a.) 𝑓(𝑥) = 𝑥 3 𝑜𝑛 [0,1] (b.) 𝑓(𝑥) =𝑠𝑖𝑛 𝑠𝑖𝑛 𝑥 𝑜𝑛 [0, −1]

Answer 1

a) [0,1]

Left sum:

$\Delta x=(a+b)/n=(0+1)/4=0.25$

Sum=0.25(f(0)+f(0.25)+f(0.5)+f(0.75))=0.25(0+0.65+0.84+0.65)=0.53

Right sum:

$\Delta x=(a+b)/n=(0+1)/4=0.25$

Sum=0.25(f(0.25)+f(0.5)+f(0.75)+f(1))=0.25(0.65+0.84+0.65+0)=0.53

Middle sum:

$\Delta x=(a+b)/n=(0+1)/4=0.25$

Sum=0.25(f(0.125)+f(0.375)+f(0.625)+f(0.875))=0.25(0.37+0.8+0.8+0.33)=0.59

b) [0,-1]

Left sum:

$\Delta x=(a+b)/n=(0-1)/4=-0.25$

Sum=-0.25(f(0)+f(-0.25)+f(-0.5)+f(-0.75))=-0.25(0-0.65-0.84-0.65)=0.53

Right sum:

$\Delta x=(a+b)/n=(0-1)/4=-0.25$

Sum=-0.25(f(-0.25)+f(-0.5)+f(-0.75)+f(-1))=-0.25(-0.65-0.84-0.65-0)=0.53

Middle sum:

$\Delta x=(a+b)/n=(0-1)/4=-0.25$

Sum=-0.25(f(-0.125)+f(-0.375)+f(-0.625)+f(-0.875))=-0.25(-0.37-0.8-0.8-0.33)=0.59