Approximate the area under the curve y=x^3+3 from x=0 to x=2 using a left sum with 4 subdivisions.

Answer 1

y = x^3 +3

exact area under the curve is the integral of x^3 + 3 = x^4/4 + 3x

evaluated from x= 0 to 2

= 2^4/4 +3(2) = 4+6 = 10

4 subdivisions from 0 to 2 each have a base length of 2/4 = 1/2

using the left side, the heights are f(0), f(1/2), f(1,), f(3/2)

= 0+3 + (1/2)^3+3 + 1^3+3 +(3/2)^3 +3

=3 + 3.125 + 4 + 6.375

=17

base times heights = 1/2 x 17 = 8.5

which is slightly less than the actual area of 10

left sums give an approximation less than the actual area

right sums give an approximation greater than the actual area

right sums with 4 subdivisions would be (1/2)(3 1/8 + 4 + 6 3/8 + 11)=12.5>10

using midpoint sums area approximation = 9.875= very close to 10