Calculus Help!!

Question

a) Evaluate the Rieman sum for f(x) = 5x^2-2x, taking the sample points to be right endpoint and a=0,b=4, and n=2

b) Evaluate (an integral of f 0 at the bottom and 4 at the top) 5x^2-2xdx, Using your result from the previous part. Then, verify your answer is correct by calculating the value of the integral using another method

Answer 1

integral of (5x^2 -2x)dx

=5(x^3/3) -2(x^2/2)

= (5/3)x^3 -x^2

evaluated from 0 to 4

= (5/3)4^3 - 4^2 -0

= (5/3)64 -16

= 320/3 -48/3

=(320-48)/3

= 272/3

=90 2/3

= area under the given parabola from x=0 to 4

Rieman sum with n= 2 using right end points

= sum of bases time heights

= 2(f(2)) + 4(f(4))

=2(5(2^2)+2(2)) + 2(5(4^2)+2(4))

=2(24) + 2(88)

= 2(112)

= 224

using left end points

= 2(f(0))+ 2(f(2))

= 0 + 48

=48

(224+48)/3 = 272/3 = 90 2/3

Rieman sum using midpoints

= 2(f(1)) + 2(f(3))

= 2(5-2) +2(5(9)-2(3))

=6 + 2(45-6)

=6+2(39)

=6+78

=84

no guarantees these calculations are error free

right end point Rieman sum should be larger than actual integral value

left end point smaller

midpoint closest to the actual integral value