Calculus Help

Name: Worksheet: Area Under a Curve - Part II Section:

1. Determine the left-hand sum for the following “curve” over the domain x�[0, 4] for each of the following n.

4

4

8

2. n = 2

3. n = 4

4. n = 8

5. Determine the exact area under the “curve”.

For use with OpenStax Calculus, free at https://openstax.org/details/books/calculus-volume-1 1

Name: Worksheet: Area Under a Curve - Part II Section:

6. Determine the exact area under the curve for each given interval. (Note that when we use the phrase ”under the curve”, curves under the x-axis measure as negative areas.)

(a) x�[−2, 4]

−4 −2 2 4

−4

−2

2

4

(b) x�[−4, 4]

−4 −2 2 4

−40

−20

20

40

(c) x�[−4, 2]

−4 −2 2 4

−4

−2

2

4

7. Consider the function f (x) = 4x3 − 6x2 − 8x + 40. Use the following DESMOS tool to help you calculate the left sums over the interval [0, 1] for the following values of n. (Note that in the tool, due to DESMOS’ syntax rules, a is used in place of n for the number of divisions.)

(a) n = 1 (a = 1) (b) n = 2 (a = 2) (c) n = 4 (a = 4) (d) n = 8 (a = 8) (e) n = 16 (a = 16)

(f) Make an estimate of the exact area under the curve on the interval [0, 1] (You can manually enter even larger numbers for a in the DESMOS tool if you want to see calculations for sums with more divisions.)

For use with OpenStax Calculus, free at https://openstax.org/details/books/calculus-volume-1 2