Question

for the function f(x)=x^3+2x, approximate the area between the curve and the x-axis over the interval [2, 0] using four equal subintervals and
a) a left-hand Riemann sum
b) a right-hand Riemann sum

Answer 1

ok we can do this but first of all there is no such interval is [2,0]

Answer 2

i assume it is [0,2]

Answer 3

lol, typo :]

Answer 4

ok so we divide the interval into 4 equal parts. lets do a) first

Answer 5

[0,2] has length 2, so dividing it into 4 equal parts means they will have length 1/2

Answer 6

clear yes?

Answer 7

yes

Answer 8

wait, length or width?

Answer 9

so we get
0, .5, 1, 1.5, 2 as the numbers for the partition

Answer 10

width i suppose since we are on the x - axis so don't let me confuse you

Answer 11

so now it is easy. you calculate base times height for each of the 4 rectangles. in part a) we use the left hand endpoints, which are 0 , .5, 1, 1.5 and compute
\[f(0)\times \frac{1}{2}+f(.5)\times \frac{1}{2}+f(1)\times \frac{1}{2} +f(1.5)\times \frac{1}{2}\]
\[=(f(0)+f(.5)+f(1)+f(1.5))\frac{1}{2}\]

Answer 12

on part b) use the right hand endpoints to get
\[(f(.5)+f(1)+f(1.5)+f(2))\times \frac{1}{2}\]

Answer 13

i would use a calculator myself. or maple if i could access it from this operating system. you want me to check the first answer?

Answer 14

5.25 for the first one?

Answer 15

and 11.25 for the second?

Answer 16

yeah i got 5.25 for the first one too