MATH 1230 Worksheet 10 – area, average, IBP, improper
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MATH 1230 Worksheet 10 – area, average, IBP, improper
1. Consider the area of the region bounded by the curves = √ , = 0 and = 4. Set up an integral that represents the area of the region. Click for a sketch.
2. Consider the area of the region bounded by the curves = √ and = . Set up an integral that represents
the area of the region. Click for a sketch.
3. Find the area of the region bounded the curves = 2 and = 0 between = and = 1. Click for a sketch.
4. Find the area of the region bounded the curves = 2 − 2 and = .Click for a sketch.
5. Find the area of the region bounded the curves = 0 and = 3 − . Click for a sketch.
6. Find the area of the region bounded the curves = ln(), = 1 and = 1. Click for a sketch. Hint: requires IBP
7. Find the area of the region bounded the curves = 62 − 31 and = 5 − 32 . Click for a sketch.
8. Find the area of the region bounded by = and = 6 − . See sketch.
9. Find the area of the region bounded by = and = − 2 .Click for a sketch.
10. Evaluate the following integrals:
(a) ∫ 2 (e) ∫ ln (h) ∫ ln()
(i) ∫ 5 3 [hint: substitute first]
(c) 5
(j) ∫ [hint: substitute first]
(d) ∫ (ln())2 (k) ∫ (Hint: use IPB with = ( − 1) and = − 2 )
1
(m) ∫
11. Evaluate the following improper integrals:
(a)
(g) −
(j) ∞ 3 − 5||
(b)
(e) − 5
(h) ∞
(k) ∞ 3 − 5||
(c)
(f) ∞ 5
(i) ∞ − 2
12. Determine the average value of () = from x = 1 to x = 5.
13. Determine () if () = {3
14. Determine () if () = {2() for ≥ 5 where () = 10
15. Determine () if () = { where () = 10
16. Determine () if () = { where () = 100
17. Determine the value of k if () = 25 for () = { for ≥ 5
18. Find the area bounded by the curves y=f(x) and y=0, for all x>0 if () = { for < 2 for ≥ 2
19. Find the area bounded by the curves 8 − 2 for ≥ ln(2)
Hint: See sketch below
20. Find the average value of () = from x=1 to x=2
21. Find the average value of () = from x=1 to x=e
22a. Find the average value of f(x) from x = 3 to x = 10 if () = ∫ () and F(10) = 21+C and F(3) = 7+C
22b. Find the average value of f(x) from x = -3 to x = 3 if () = ∫ () and F(10) = 24+C and F(3) = 48+C
23. Find the average value of f(x) from x = -ln(2) to x = ln(2) if () = 16 − ||
24. Find the average value of f(x) from x = -ln(4) to x = ln(2) if () = 16 ||
25. Find the average value of f(x) from x = -2 to x = 4 if () = {
26. Find the average value of f(x) from x = -1 to x = 2 if () = where () = 10
27. If the average value of f(x) from x= 5 to x = 10 is 100, determine ∫()
28. If the average value of f(x) from x= -10 to x = 10 is 100, determine 0 ()
29. If the average value of f(x) from = − 1000 to = 1000 is 0, determine ()
30. If the average value of f(x) from = 0 to = is 100, determine ()
31. If the average value of f(x)=2x from = 0 to = is 100, determine ℎ ()
32. If the average value of f(x)=3x2 from = 0 to = is 100, determine ℎ ()
The below are extra/optional questions, meant to prompt some thinking and discussion.
Extra 1. If the average value of f(x) from = − ∞ to = ∞ is 0, what can you say about ∞ () ??
Extra 2. If the average value of f(x) from = − ∞ to = ∞ is 100, what can you say about ∞ () ??
Extra 3. If the average value of f(x) from = − ∞ to = ∞ is ∞, what can you say about ∞ () ??
2022-03-31