Approximate area under a curve

Answer Key - Grade 12 Math Worksheet - Challenge Me

Circle the best answer, or write your answer on the line.

1. 1.

   Esme's function f rises all the way from x = 0 to x = 4. Is a left sum for the area too big or too small?

   • It is an underestimate: the left ends are the lowest points of each strip
   • It is an overestimate: the left ends are the highest points of each strip
   • It is exact: each rectangle touches the curve at a corner
2. 2.

   Jun's function f falls all the way from x = 1 to x = 5. Is a right sum for the area too big or too small?

   • It is an underestimate: the right ends are the lowest points of each strip
   • It is an overestimate: the right ends are the highest points of each strip
   • It is exact: each rectangle touches the curve at a corner
3. 3.

   Kira's function f rises all the way from x = 2 to x = 8. Is a right sum for the area too big or too small?

   • It is exact: each rectangle touches the curve at a corner
   • It is an overestimate: the right ends are the highest points of each strip
   • It is an underestimate: the right ends are the lowest points of each strip

4. Read this first

   Speed of the bike (example data)

   time (s)	speed (m/s)
   0	2
   5	4
   15	5
   30	7

5. 4.

   Milo records the speed of a toy bike at uneven times (see the table). Estimate the distance in meters with a left sum. What is it?

   125

6. Speed of the kite (example data)

   time (s)	speed (m/s)
   0	1
   2	3
   6	4
   12	6

7. 5.

   Ren records the speed of a toy kite at uneven times (see the table). Estimate the distance in meters with a left sum. What is it?

   38

8. Speed of the car (example data)

   time (s)	speed (m/s)
   0	5
   20	8
   30	10
   60	12

9. 6.

   Zee records the speed of a toy car at uneven times (see the table). Estimate the distance in meters with a left sum. What is it?

   480
10. 7.

    Tala's rising function has f(1) = 1 and f(5) = 25. With 8 equal strips on that interval, how much bigger is the right sum than the left sum?

    12
11. 8.

    Pia's rising function has f(0) = 2 and f(6) = 20. With 3 equal strips on that interval, how much bigger is the right sum than the left sum?

    36
12. 9.

    Bo's rising function has f(2) = 5 and f(4) = 13. With 5 equal strips on that interval, how much bigger is the right sum than the left sum?

    3.2
13. 10.

    Ari's rising function has f(−2) = 1 and f(4) = 10. With 3 equal strips on that interval, how much bigger is the right sum than the left sum?

    18