5.E: Applications of Integration (Exercises)

5.1: Antiderivatives and Indefinite Integration

Terms and Concepts

1. Define the term “antiderivative” in your own words

2. Is it more accurate to refer to “the” antiderivative of \(f(x)\) or “an” antiderivative of \(f(x)\)?

3. Use your own words to define the indefinite integral of \(f(x)\).

4. Fill in the blanks: “Inverse operations do the ____ things in the _____ order.”

5. What is an “initial value problem”?

6. The derivative of a position function is a _____ function.

7. The antiderivative of an acceleration function is a ______ function.

Problems

In Exercises 8-26, evaluate the given indefinite integral.

8. \(\int 3x^3 \,dx\)

9. \(\int x^8 \,dx\)

10. \(\int (10x^2-2) \,dx\)

11. \(\int \,dt\)

12. \(\int 1 \,ds\)

13. \(\int \frac{1}{3t^2}\, dt\)

14. \(\int \frac{1}{t^2}\, dt\)

15. \(\int \frac{1}{\sqrt{x}}\, dx\)

16. \(\int \sec^2 \theta\, d\theta\)

17. \(\int \sin \theta\, d\theta\)

18. \(\int (\sec x \tan x +\csc x \cot x )\, dx\)

19. \(\int 5e^\theta\, d\theta\)

20. \(\int 3^t\, dt\)

21. \(\int \frac{5^t}{2}\, dt\)

22. \(\int (2t+3)^2\, dt\)

23. \(\int (t^2+3)(t^3-2t)\, dt\)

24. \(\int x^2x^3\, dx\)

25. \(\int e^\pi\, dx\)

26. \(\int a\, dx\)

27. This problem investigates why Theorem 35 states that \(\int \frac{1}{x}\,dx = \ln |x|+C\).
(a) What is the domain of \(y=\ln x\)?
(b) Find \(\frac{d}{dx}(\ln x)\).
(c) What is the domain of \(y=\ln (-x)\)?
(d) Find \(\frac{d}{dx}\left ( (\ln (-x)\right )\).
(e) You should find that \(1/x\) has two types of antiderivatives, depending on whether \(x>0\) or \(x<0\). In one expression, give a formula for \(\int \frac{1}{x}\,dx\) that takes these different domains into account, and explain your answer.

In Exercises 28-38, find \(f(x)\) described by the given initial value problem.

28. \(f'(x)=\sin x\text{ and }f(0)=2\)

29. \(f'(x)=5e^x\text{ and }f(0)=10\)

30. \(f'(x)=4x^3-3x^2\text{ and }f(-1)=9\)

31. \(f'(x)=\sec^2 x\text{ and }f(\pi/4)=5\)

32. \(f'(x)=7^x\text{ and }f(2)=1\)

33. \(f''(x)=5\text{ and }f'(0)=7,f(0)=3\)

34. \(f''(x)=7x\text{ and }f'(1)=-1,f(1)=10\)

35. \(f''(x)=5e^x\text{ and }f'(0)=3,f(0)=5\)

36. \(f''(\theta)=\sin \theta \text{ and }f'(\pi)=2,f(\pi)=4\)

37. \(f''(x)=24x^2+2^x-\cos x \text{ and }f'(0)=5,f(0)=0\)

38. \(f''(x)=0\text{ and }f'(1)=3,f(1)=1\)

Review

39. Use information gained from the first and second derivative to sketch \(f(x)=\frac{1}{e^x+1}\).

40. Given \(y=x^2e^x\cos x\), find \(dy\).

5.2: The Definite Integral

Terms and Concepts

1. What is "total signed area"?

2. What is "displacement"?

3. What is \(\int_3^3 \sin x\,dx\)

4. Give a single definite integral that has the same value as \(\int_0^1 (2x+3)\,dx +\int_1^2 (2x+3)\,dx\).

Problems

In Exercises 5-9, a graph of a function \(f(x)\) is given. Using the geometry of the graph, evaluate the definite integrals.

5.

(a) \(\int_0^1 (-2x+4)\,dx\)
(b) \(\int_0^2 (-2x+4)\,dx\)
(c) \(\int_0^3 (-2x+4)\,dx\)
(d) \(\int_1^3 (-2x+4)\,dx\)
(e) \(\int_2^4 (-2x+4)\,dx\)
(f) \(\int_0^1 (-6x+12)\,dx\)

6.

(a) \(\int_0^2 f(x)\,dx\)
(b) \(\int_0^3 f(x)\,dx\)
(c) \(\int_0^5 f(x)\,dx\)
(d) \(\int_2^5 f(x)\,dx\)
(e) \(\int_5^3 f(x)\,dx\)
(f) \(\int_0^3 f(x)\,dx\)

7.

(a) \(\int_0^2 f(x)\,dx\)
(b) \(\int_2^4 f(x)\,dx\)
(c) \(\int_2^4 2f(x)\,dx\)
(d) \(\int_0^1 4x\,dx\)
(e) \(\int_2^3 (2x-4)\,dx\)
(f) \(\int_2^3 (4x-8)\,dx\)

8.

(a) \(\int_0^1 (x-1)\,dx\)
(b) \(\int_0^2 (x-1)\,dx\)
(c) \(\int_0^3 (x-1)\,dx\)
(d) \(\int_2^3 (x-1)\,dx\)
(e) \(\int_1^4 (x-1)\,dx\)
(f) \(\int_1^4 \left ((x-1)+1\right )\,dx\)

9.

(a) \(\int_0^2 f(x)\,dx\)
(b) \(\int_2^4 f(x)\,dx\)
(c) \(\int_0^4 f(x)\,dx\)
(d) \(\int_0^4 5f(x)\,dx\)

In Exercises 10-13, a graph of a function \(f(x)\) is given; the numbers inside the shaded regions give the area of that region. Evaluate the definite integrals using this area information.

10.

(a) \(\int_0^1 f(x)\,dx\)
(b) \(\int_0^2 f(x)\,dx\)
(c) \(\int_0^3 f(x)\,dx\)
(d) \(\int_1^2 -3f(x)\,dx\)

11.

(a) \(\int_0^2 f(x)\, dx\)
(b) \(\int_2^4 f(x)\, dx\)
(c) \(\int_0^4 f(x)\, dx\)
(d) \(\int_0^1 f(x)\, dx\)

12.

(a) \(\int_{-2}^{-1}f(x)\,dx\)
(b) \(\int_{1}^{2}f(x)\,dx\)
(c) \(\int_{-1}^{1}f(x)\,dx\)
(d) \(\int_{0}^{1}f(x)\,dx\)

13.

(a) \(\int_0^2 5x^2\,dx\)
(b) \(\int_0^2 (x^2+1)\,dx\)
(c) \(\int_1^3 (x-1)^2\,dx\)
(d) \(\int_2^4 \left ( (x-2)+5\right )\,dx\)

In Exercises 14-15, a graph of the velocity function of an object moving in a straight line is given. Answer the questions based on that graph.

14.

(a) What is the object's maximum velocity?
(b) What is the object's maximum displacement?
(c) What is the object's total displacement on [0,3]?

15.

(a) What is the object's maximum velocity?
(b) What is the object's maximum displacement?
(c) What is the object's total displacement on [0,5]?

16. An object is thrown straight up with a velocity, in ft/s, given by \(v(t) = -32t+64\), where \(t\) is in seconds, from a height of 48 feet.
(a) What is the object's maximum velocity?
(b) What is the object's maximum displacement?
(c) When does the maximum displacement occur?
(d) When will the object reach a height of 0? (Hint: find when the displacement is -48ft.)

17. An object is thrown straight up with a velocity, in ft/s, given by \(v(t)=-32t+96\), where \(t\) is seconds, from a height of 64 feet.
(a) What is the object's initial velocity?
(b) What is the object's displacement 0?
(c) How long does it take for the object to return to its initial height?
(d) When will the object reach a height of 210ft?

In Exercises 18-21, let

• \(\int_0^2 f(x) \,dx=5\),
• \(\int_0^3 f(x) \,dx=7\),
• \(\int_0^2 g(x) \,dx=-3\), and
• \(\int_2^3 g(x) \,dx=5\).

Use these values to evaluate the given definite integrals.

18. \(\int_0^2 \left ( f(x)+g(x)\right )\,dx\)

19. \(\int_0^3 \left ( f(x)-g(x)\right )\,dx\)

20. \(\int_2^3 \left ( 3f(x)+2g(x)\right )\,dx\)

21. Find values for \(a\) and \(b\) such that
\(\int_0^3 \left ( af(x)+bg(x)\right )\,dx=0\)

In Exercises 22-25, let

• \(\int_0^3 s(t)\,dt =10\),
• \(\int_3^5 s(t)\,dt =8\),
• \(\int_3^5 r(t)\,dt =-1\), and
• \(\int_0^5 r(t)\,dt =11\).

Use these values to evaluate the given definite integrals.

22. \(\int_0^3 \left ( s(t)+r(t)\right )\,dt\)

23. \(\int_5^0 \left ( s(t)-r(t)\right )\,dt\)

24. \(\int_3^3 \left ( \pi s(t)-7r(t)\right )\,dt\)

25. Find values for a and b such that
\(\int_0^5 \left ( ar(t)+bs(t)\right )\,dt=0\)

Review

In Exercises 26-29, evaluate the given indefinite integral.

26. \(\int (x^3-2x^2+7x-9)\,dx\)

27. \(\int (\sin x -\cos x +\sec^2 x)\,dx\)

28. \(\int (\sqrt[3]{t}+\frac{1}{t^2}+2^t)\,dt\)

29. \(\int \left ( \frac{1}{x} -\csc x \cot x \right )\,dx\)

5.3: Riemann Sums

Terms and Concepts

1. A fundamental calculus technique is to use ________ to refine approximations to get an exact answer.

2. What is the upper bound in the summation \(\sum_{i=7}^{14} (48i-201)\)?

3. This section approximates definite integrals using what geometric shape?

4. T/F: A sum using the Right Hand Rule is an example of a Riemann Sum.

Problems

In Exercises 5-11, write out each term of the summation and compute the sum.

5. \(\sum_{i=2}^4 i^2\)

6. \(\sum_{i=-1}^3 (4i-2)\)

7. \(\sum_{i=-2}^2 \sin (\pi i/2)\)

8. \(\sum_{i=1}^5 \frac{1}{i}\)

9. \(\sum_{i=1}^6 (-1)^i i\)

10. \(\sum_{i=1}^4 \left ( \frac{1}{i}-\frac{1}{i+1}\right )\)

11. \(\sum_{i=0}^5 (-1)^i \cos (\pi i)\)

In Exercises 12-15, write each sum in summation notation.

12. \(3+6+9+12+15\)

13. \(-1+0+3+8+15+24+35+48+63\)

14. \(\frac{1}{2}+\frac{2}{3}+\frac{3}{4}+\frac{4}{5}\)

15. \(1-e+e^2-e^3+e^4\)

In Exercises 16-22, evaluate the summation using Theorem 37.

16. \(\sum_{i=1}^25 i\)

17. \(\sum_{i=1}^10 (3i^2-2i)\)

18. \(\sum_{i=1}^15 (2i^3-10)\)

19. \(\sum_{i=1}^10 (-4i^3+10i^2-7i+11)\)

20. \(\sum_{i=1}^10 (i^3-3t^2+2i+7)\)

21. \(1+2+3+... + 99+100\)

22. \(1+4+9+...+361+400\)

Theorem 37 states

\(\sum_{i=1}^n a_i = \sum+{i=1}^k a_i+\sum_{i=k+1}^n a_i\), so

\(\sum_{i=k+1}^n a_i = \sum_{i=1}^n a_i -\sum_{i=1}^k a_i \).

Use this fact, along with other parts of Theorem 37, to evaluate the summations given in Exercises 23-26.

23. \(\sum_{i=11}^20 i\)

24. \(\sum_{i=16}^25 i^3\)

25. \(\sum_{i=7}^12 4\)

26. \(\sum_{i=5}^10 4i^3\)

In Exercises 27-32, a definite integral
\(\int_a^b f(x)\,dx \) is given.
(a) Graph \(f(x)\) on [a,b].
(b) Add to the sketch rectangles using the provided rule.
(c) Approximate \(\int_a^b f(x)\,dx\) by summing the areas of the rectangles.

27. \(\int_{-3}^3 x^2\,dx\), with 6 rectangles using the Left Hand Rule.

28. \(\int_{0}^2 (5-x^2)\,dx\), with 4 rectangles using the Midpoint Rule.

29. \(\int_0^{\pi}\sin x\,dx\), with 6 rectangles using the Right Hand Rule.

30. \(\int_0^3 2^x\,dx\), with 5 rectangles using the Left Hand Rule.

31. \(\int_1^2 \ln x\,dx\), with 3 rectangles using the Midpoint Rule.

32. \(\int_1^9 \frac{1}{x} \,dx\), with 4 rectangles using the Right Hand Rule.

In Exercises 33-38, a definite integral
\(\int_a^b f(x)\,dx\) is given. As demonstrated in Example 123 and 124, do the following.
(a) Find a formula to approximate \(\int_a^b f(x)\,dx\) using \(n\) subintervals and the provided rule.
(b) Evaluate the formula using \(n=10,\,100\text{ and }1000.\)
(c) Find the limit of the formula, as \(n\to \infty\), to find the exact value of \(\int_a^b f(x)\,dx\).

33. \(\int_0^1 x^3\,dx\), using the Right Hand Rule.

34. \(\int_{-1}^1 3x^2\,dx\), using the Left Hand Rule.

35. \(\int_{-1}^3 (3x-1)\,dx\), using the Midpoint Rule.

36. \(\int_1^4 (2x^2-3)\,dx\), using the Left Hand Rule.

37. \(\int_{-10}^{10}(5-x)\,dx\), using the Right Hand Rule.

38. \(\int_0^1 (x^3-x^2)\,dx\), using the Right Hand Rule.

Review

In Exercises 39-44, find an antiderivative of the given function.

39. \(f(x) = 5\sec^2 x\)

40. \(f(x) = \frac{7}{x}\)

41. \(g(t) = 4t^5-5t^3+8\)

42. \(g(t) =5\cdot 8^t\)

43. \(g(t) =\cos t +\sin t\)

44. \(f(x) =\frac{1}{\sqrt{x}}\)

5.4: The Fundamental Theorem of Calculus

Terms and Concepts

1. How are the definite and indefinite integrals related?

2. What constants of integration is most commonly used when evaluating definite integrals?

3. T/F: If \(f\) is a continuous function, then \(F(x) =\int_a^x f(t)\,dt\) is also a continuous function.

4. The definite integral can be used to find "the area under a curve." Give two other uses for definite integrals.

Problems

In Exercises 5-28, evaluate the definite integral.

5. \(\int_1^3 (3x^2-2x+1)\,dx\)

6. \(\int_0^4 (x-1)^2\,dx\)

7. \(\int_{-1}^1 (x^3-x^5)\,dx\)

8. \(\int_{\pi/2}^{\pi}\cos x\,dx\)

9. \(\int_0^{\pi/4}\sec^2 x\,dx\)

10. \(\int_1^e \frac{1}{x}\,dx\)

11. \(\int_{-1}^1 5^x \,dx\)

12. \(\int_{-2}^{-1}(4-2x^3)\,dx\)

13. \(\int_0^{\pi}(2\cos x -2\sin x)\,dx\)

14. \(\int_1^3 e^x\,dx\)

15. \(\int_0^4 \sqrt{t}\,dt\)

16. \(\int_9^{25} \frac{1}{\sqrt{t}}\,dt\)

17. \(\int_1^8 \sqrt[3]{x}\,dx\)

18. \(\int_1^2 \frac{1}{x}\,dx\)

19. \(\int_1^2 \frac{1}{x^2}\,dx\)

20. \(\int_1^2 \frac{1}{x^3}\,dx\)

21. \(\int_0^1 x\,dx\)

22. \(\int_0^1 x^2\,dx\)

23. \(\int_0^1 x^3\,dx\)

24. \(\int_0^1 x^{100}\,dx\)

25. \(\int_{-4}^4 dx\)

26. \(\int_{-10}^{-5} 3\,dx\)

27. \(\int_{-2}^2 0\,dx\)

28. \(\int_{\pi/6}^{\pi/3}\csc x \cot x\,dx\)

29. Explain why:
(a) \(\int_{-1}^1 x^n\,dx=0\), when n is a positive, odd integer, and
(b) \(\int_{-1}^1x^n\,dx =2\int_0^1 x^n \,dx\) when n is a positive, even integer.

In Exercises 30-33, find a value c guaranteed by the Mean Value Theorem.

30. \(\int_0^2 x^2\,dx\)

31. \(\int_{-2}^2 x^2\,dx\)

32. \(\int_0^1 e^x\,dx\)

33 \(\int_0^16 \sqrt{x}\,dx\)

In Exercises 34-39, find the average value of the function on the given interval.

34. \(f(x) =\sin x \text{ on }[0,\pi/2]\)

35. \(y =\sin x \text{ on }[0,\pi]\)

36. \(y = x \text{ on }[0,4]\)

37. \(y =x^2 \text{ on }[0,4]\)

38. \(y =x^3 \text{ on }[0,4]\)

39. \(g(t) =1/t \text{ on }[1,e]\)

In Exercises 40-44, a velocity function of an object moving along a straight line is given. Find the displacement of the object over the given time interval.

40. \(v(t) =-32t+20\)ft/s on [0,5].

41. \(v(t) =-32t+200\)ft/s on [0,10].

42. \(v(t) =2^t\)mph on [-1,1].

43. \(v(t) =\cos t\)ft/s on \([0,3\pi /2]\).

44. \(v(t) =\sqrt[4]{t}\)ft/s on [0,16].

In Exercises 45-48, an acceleration function of an object moving along a straight line is given. Find the change of the object's velocity over the given time interval.

45. \(a(t) =-32\)ft/s on [0,2].

46. \(a(t) =10\)ft/s on [0,5].

47. \(a(t) =t\)ft/s\(^2\) on [0,2].

48. \(a(t) =\cos t\)ft/s\(^2\) on \([0,\pi]\).

In Exercises 49-52, sketch the given functions and fine the area of the enclosed region.

49. \(y =2x,\, y=5x,\text{ and }x=3\).

50. \(y=-x+1,\,y=3x+6,\,x=2\text{ and }x=-1\).

51. \(y=x^2-2x+5,\,y=5x-5\).

52. \(y = 2x^2+2x-5,\,y=x^2+3x+7\).

In Exercises 53-56, find \(F'(x)\).

53. \(F(x) =\int_2^{x^3+x}\frac{1}{t}\,dt\)

54. \(F(x) = \int_{x^3}^0 t^3\,dt\)

55. \(F(x)=\frac{x}{x^2}(t+2)\,dt\)

56. \(F(x) =\int_{\ln x}^{e^x}\sin t\,dt\)

5.5: Numerical Integration

Terms and Concepts

1. T/F: Simpson's Rule is a method of approximating antiderivatives.

2. What are the two basic situations where approximating the value of a definite integral is necessary?

3. Why are the Left and Right Hand Rules rarely used?

Problems

In Exercises 4-11, a definite integral is given.
(a) Approximating the definite integral with the Trapezoidal Rule and \(n=4\).
(b) Approximate the definite integral with Simpson's Rule and \(n=4\).
(c) Find the exact value of the integral.

4. \(\int_{-1}^1 x^2\,dx\)

5. \(\int_{0}^{10} 5x\,dx\)

6. \(\int_{0}^{\pi} \sin x\,dx\)

7. \(\int_{0}^{4} \sqrt{x}\,dx\)

8. \(\int_{0}^{3} (x^3+2x^2-5x+7)\,dx\)

9. \(\int_{0}^{1} x^4\,dx\)

10. \(\int_{0}^{2x} \cos x\,dx\)

11. \(\int_{-3}^{3} \sqrt{9-x^2}\,dx\)

In Exercises 12-19, approximate the definite integral with the Trapezoidal Rule and Simpson's Rule, with \(n=6\).

12. \(\int_{0}^{1}\cos (x^2)\,dx\)

13. \(\int_{-1}^{1}e^{x^2}\,dx\)

14. \(\int_{0}^{5}\sqrt{x^2+1}\,dx\)

15. \(\int_{0}^{\pi}x\sin x\,dx\)

16. \(\int_{0}^{\pi/2}\sqrt{\cos x}\,dx\)

17. \(\int_{1}^{4}\ln x\,dx\)

18. \(\int_{-1}^{1}\frac{1}{\sin x +2}\,dx\)

19. \(\int_{0}^{6}\frac{1}{\sin x +2}\,dx\)

In Exercises 20-23, find n such that the error in approximating the given definite integral is less than 0.0001 when using:
(a) the Trapezoidal Rule
(b) Simpson's Rule

20. \(\int_{0}^{\pi} \sin x\,dx\)

21. \(\int_{1}^{4}\frac{1}{\sqrt{x}}\,dx\)

22. \(\int_{0}^{\pi} \cos (x^2)\,dx\)

23. \(\int_{0}^{5} x^4\,dx\)

In Exercises 24-25, a region is given. Find the area of the region using Simpson's Rule:
(a) where the measurements are in centimeters, taken in 1cm increments, and
(b) where the measurements are in hundreds of yards, taken in 100 yd increments.

24.

25.