4.3.2: Homework

Reading Questions

1. How is the definite integral \( \displaystyle \int_a^b f(x) \, dx\) defined in terms of a limit of Riemann sums?
2. What are the "limits of integration" in a definite integral? What do \(a\) and \(b\) represent?
3. What is the difference between a definite integral and an indefinite integral?
4. What does it mean for a function to be "integrable" on an interval \([a,b]\)? Are all continuous functions on \([a,b]\) integrable?
5. How does the definite integral relate to the concept of "net signed area"? Explain what happens to areas below the \( x \)-axis.
6. How can you find the "total area" between a curve \(y=f(x)\) and the \( x \)-axis over \([a,b]\) using a definite integral?
7. If \(v(t)\) is a velocity function, what does \( \displaystyle \int_a^b v(t) \, dt\) represent?
8. State the property of definite integrals: \( \displaystyle \int_a^a f(x) \, dx = \)?
9. State the property of definite integrals: \( \displaystyle \int_a^b f(x) \, dx = ? \cdot \displaystyle \int_b^a f(x) \, dx\)
10. State the property of definite integrals regarding the integral of a sum: \( \displaystyle \int_a^b [f(x)+g(x)] \, dx = \)?
11. State the property of definite integrals: \( \displaystyle \int_a^b cf(x) \, dx = \)? (where \(c\) is a constant).
12. State the additivity property of definite integrals: \( \displaystyle \int_a^b f(x) \, dx = \displaystyle \int_a^c f(x) \, dx + \)?
13. What does the Comparison Theorem state about \( \displaystyle \int_a^b f(x) \, dx\) if \(f(x) \ge 0\) on \([a,b]\)?

Homework

In exercises 1 - 4, express the limits as integrals.

1) \(\displaystyle \lim_{n \to \infty }\sum_{i=1}^n(x^∗_i) \Delta x\) over \([1,3]\)

2) \(\displaystyle \lim_{n \to \infty }\sum_{i=1}^n(5(x^∗_i)^2−3(x^∗_i)^3) \Delta x\) over \([0,2]\)

Answer

\(\displaystyle \int^2_0(5x^2−3x^3)\,dx\)

3) \(\displaystyle \lim_{n \to \infty }\sum_{i=1}^n\sin^2(2 \pi x^∗_i) \Delta x\) over \([0,1]\)

4) \(\displaystyle \lim_{n \to \infty }\sum_{i=1}^n\cos^2(2 \pi x^∗_i) \Delta x\) over \([0,1]\)

Answer

\(\displaystyle \int^1_0\cos^2(2 \pi x)\,dx\)

In exercises 5 - 10, given \(L_n\) or \(R_n\) as indicated, express their limits as \(n \to \infty \) as definite integrals, identifying the correct intervals.

5) \(\displaystyle L_n=\frac{1}{n}\sum_{i=1}^n\frac{i−1}{n}\)

6) \(\displaystyle R_n=\frac{1}{n}\sum_{i=1}^n\frac{i}{n}\)

Answer

\(\displaystyle \int^1_0x\,dx\)

7) \(\displaystyle Ln=\frac{2}{n}\sum_{i=1}^n(1+2\frac{i−1}{n})\)

8) \(\displaystyle R_n=\frac{3}{n}\sum_{i=1}^n(3+3\frac{i}{n})\)

Answer

\(\displaystyle \int^6_3x\,dx\)

9) \(\displaystyle L_n=\frac{2 \pi }{n}\sum_{i=1}^n2 \pi \frac{i−1}{n}\cos(2 \pi \frac{i−1}{n})\)

10 \(\displaystyle R_n=\frac{1}{n}\sum_{i=1}^n(1+\frac{i}{n})\log((1+\frac{i}{n})^2)\)

Answer

\(\displaystyle \int^2_1x\log(x^2)\,dx\)

In exercises 11 - 16, evaluate the integrals of the functions graphed using the formulas for areas of triangles and circles, and subtracting the areas below the \(x\)-axis.

11)

12)

Answer

\( 1+2 \cdot 2+3 \cdot 3=14\)

13)

14)

Answer

\(1−4+9=6\)

15)

16)

Answer

\(1−2 \pi +9=10−2 \pi \)

In exercises 17 - 24, evaluate the integral using area formulas.

17) \(\displaystyle \int^3_0(3−x)\,dx\)

18) \(\displaystyle \int^3_2(3−x)\,dx\)

Answer

The integral is the area of the triangle, \(\frac{1}{2}\).

19) \(\displaystyle \int^3_{−3}(3−|x|)\,dx\)

20) \(\displaystyle \int^6_0(3−|x−3|)\,dx\)

Answer

The integral is the area of the triangle, \(9\).

21) \(\displaystyle \int^2_{−2}\sqrt{4−x^2}\,dx\)

22) \(\displaystyle \int^5_1\sqrt{4−(x−3)^2}\,dx\)

Answer

The integral is the area \(\frac{1}{2} \pi r^2=2 \pi \).

23) \(\displaystyle \int^{12}_0\sqrt{36−(x−6)^2}\,dx\)

24) \(\displaystyle \int^3_{−2}(3−|x|)\,dx\)

Answer

The integral is the area of the "big" triangle less the "missing" triangle, \(9−\frac{1}{2}\).

Suppose that \(\displaystyle \int^4_0f(x)\,dx=5\) and \(\displaystyle \int^2_0f(x)\,dx=−3\), and \(\displaystyle \int^4_0g(x)\,dx=−1\) and \(\displaystyle \int^2_0g(x)\,dx=2\). In exercises 25 - 30, compute the integrals.

25) \(\displaystyle \int^4_0(f(x)+g(x))\,dx\)

26) \(\displaystyle \int^4_2(f(x)+g(x))\,dx\)

Answer

\(\displaystyle \int^4_2f(x)\,dx+\int^4_2g(x)\,dx=8−3=5\)

27) \(\displaystyle \int^2_0(f(x)−g(x))\,dx\)

28) \(\displaystyle \int^4_2(f(x)−g(x))\,dx\)

Answer

\(\displaystyle \int^4_2f(x)\,dx−\int^4_2g(x)\,dx=8+3=11\)

29) \(\displaystyle \int^2_0(3f(x)−4g(x))\,dx\)

30) \(\displaystyle \int^4_2(4f(x)−3g(x))\,dx\)

Answer

\(\displaystyle 4\int^4_2f(x)\,dx−3\int^4_2g(x)\,dx=32+9=41\)

In exercises 31 - 34, use the identity \(\displaystyle \int^A_{−A}f(x)\,dx=\int^0_{−A}f(x)\,dx+\int^A_0f(x)\,dx\) to compute the integrals.

31) \(\displaystyle \int^ \pi _{− \pi }\frac{\sin t}{1+t^2}dt\) (Hint: \(\displaystyle \sin(−t)=−\sin(t))\)

32) \(\displaystyle \int^{\sqrt{ \pi }}_\sqrt{− \pi }\frac{t}{1+\cos t}dt\)

Answer

The integrand is odd; the integral is zero.

33) \(\displaystyle \int^3_1(2−x)\,dx\) (Hint: Look at the graph of \(f\).)

34) \(\displaystyle \int^4_2(x−3)^3\,dx\) (Hint: Look at the graph of \(f\).)

Answer

The integrand is antisymmetric with respect to \(x=3\). The integral is zero.

In exercises 35 - 40, given that \(\displaystyle \int^1_0x\,dx=\frac{1}{2},\;\int^1_0x^2\,dx=\frac{1}{3}\), and \(\displaystyle \int^1_0x^3\,dx=\frac{1}{4}\), compute the integrals.

35) \(\displaystyle \int^1_0(1+x+x^2+x^3)\,dx\)

36) \(\displaystyle \int^1_0(1−x+x^2−x^3)\,dx\)

Answer

\(\displaystyle 1−\frac{1}{2}+\frac{1}{3}−\frac{1}{4}=\frac{7}{12}\)

37) \(\displaystyle \int^1_0(1−x)^2\,dx\)

38) \(\displaystyle \int^1_0(1−2x)^3\,dx\)

Answer

\(\displaystyle \int^1_0(1−6x+12x^2−8x^3)\,dx = 1 −6\left( \frac{1}{2} \right) \) \(+12\left(\frac{1}{3}\right)−8\left(\frac{1}{4}\right)=1-3+4-2 \) \(=0\)

39) \(\displaystyle \int^1_0\left(6x−\tfrac{4}{3}x^2\right)\,dx\)

40) \(\displaystyle \int^1_0(7−5x^3)\,dx\)

Answer

\(7−\frac{5}{4}=\frac{23}{4}\)

In exercises 41 - 46, use the comparison theorem.

41) Show that \(\displaystyle \int^3_0(x^2−6x+9)\,dx \geq 0\).

42) Show that \(\displaystyle \int^3_{−2}(x−3)(x+2)\,dx \leq 0\).

Answer

The integrand is negative over \([−2,3]\).

43) Show that \(\displaystyle \int^1_0\sqrt{1+x^3}\,dx \leq \int^1_0\sqrt{1+x^2}\,dx\).

44) Show that \(\displaystyle \int^2_1\sqrt{1+x}\,dx \leq \int^2_1\sqrt{1+x^2}\,dx\).

Answer

\(x \leq x^2\) over \([1,2]\), so \(\sqrt{1+x} \leq \sqrt{1+x^2}\) over \([1,2]\).

45) Show that \(\displaystyle \int^{ \pi /2}_0\sin tdt \geq \frac{ \pi }{4}\) (Hint: \(\sin t \geq \frac{2t}{ \pi }\) over \( [0,\frac{ \pi }{2}])\)

46) Show that \(\displaystyle \int^{ \pi /4}_{− \pi /4}\cos t\,dt \geq \pi \sqrt{2}/4\).

Answer

\(\cos(t) \geq \dfrac{\sqrt{2}}{2}\). Multiply by the length of the interval to get the inequality.

47) Suppose that \(\displaystyle A=\int^{2 \pi }_0\sin^2t\,dt\) and \(\displaystyle B=\int^{2 \pi }_0\cos^2t\,dt\). Show that \(A+B=2 \pi \) and \(A=B\).

48) Suppose that \(\displaystyle A=\int^{ \pi /4}_{− \pi /4}\sec^2 t\,dt= \pi \) and \(\displaystyle B=\int^{ \pi /4}_{− \pi /}4\tan^2 t\,dt\). Show that \(A−B=\dfrac{ \pi }{2}\).

Answer

Use \(\tan^2 \theta +1=\sec^2 \theta \). Then, \(\displaystyle B−A=\int^{ \pi /4}_{− \pi /4}1\,dx=\frac{ \pi }{2}\).

49) Explain why the graphs of a quadratic function (parabola) \(p(x)\) and a linear function \(ℓ(x)\) can intersect in at most two points. Suppose that \(p(a)=ℓ(a)\) and \(p(b)=ℓ(b)\), and that \(\displaystyle \int^b_ap(t)\,dt>\int^b_aℓ(t)dt\). Explain why \(\displaystyle \int^d_cp(t)>\int^d_cℓ(t)\,dt\) whenever \( a \leq c<d \leq b\).

50) Suppose that parabola \(p(x)=ax^2+bx+c\) opens downward \((a<0)\) and has a vertex of \(y=\dfrac{−b}{2a}>0\). For which interval \([A,B]\) is \(\displaystyle \int^B_A(ax^2+bx+c)\,dx\) as large as possible?

Answer

The integral is maximized when one uses the largest interval on which \(p\) is nonnegative. Thus, \(A=\frac{−b−\sqrt{b^2−4ac}}{2a}\) and \(B=\frac{−b+\sqrt{b^2−4ac}}{2a}\).

51) Suppose \([a,b]\) can be subdivided into subintervals \(a=a_0<a_1<a_2< \cdots <a_N=b\) such that either \(f \geq 0\) over \([a_{i−1},a_i]\) or \(f \leq 0\) over \([a_{i−1},a_i]\). Set \(\displaystyle A_i=\int^{a_i}_{a_{i−1}}f(t)\,dt\).

a. Explain why \(\displaystyle \int^b_af(t)\,dt=A_1+A_2+ \cdots +A_N\).

b. Then, explain why \(\displaystyle \int^b_af(t)\,dt \leq \int^b_a|f(t)|\,dt\).

52) Suppose \(f\) and \(g\) are continuous functions such that \(\displaystyle \int^d_cf(t)\,dt \leq \int^d_cg(t)\,dt\) for every subinterval \([c,d]\) of \([a,b]\). Explain why \( f(x) \leq g(x)\) for all values of \(x\).

Answer

If \(f(t_0)>g(t_0)\) for some \(t_0 \in [a,b]\), then since \(f−g\) is continuous, there is an interval containing \(t_0\) such that \( f(t)>g(t)\) over the interval \([c,d]\), and then \(\displaystyle \int^d_df(t)\,dt>\int^d_cg(t)\,dt\)over this interval.

53) Suppose that for each \(i\) such that \( 1 \leq i \leq N\) one has \(\displaystyle \int^i_{i−1}f(t)\,dt=i\). Show that \(\displaystyle \int^N_0f(t)\,dt=\frac{N(N+1)}{2}\).

54) Suppose that for each \(i\) such that \(1 \leq i \leq N\) one has \(\displaystyle \int^i_{i−1}f(t)\,dt=i^2\). Show that \(\displaystyle \int^N_0f(t)\,dt=\frac{N(N+1)(2N+1)}{6}\).

Answer

\[ \begin{array}{rcl} \displaystyle \int^N_0f(t)\,dt & = & \displaystyle \sum_{i=1}^N\int^i_{i−1}f(t)\,dt \\[6pt] & = & \displaystyle \sum_{i=1}^Ni^2 \\[6pt] & = & \dfrac{N(N+1)(2N+1)}{6} \\[6pt] \end{array} \nonumber \]

55) If \(\displaystyle \int^5_1\sqrt{1+t^4}\,dt=41.7133...\), what is \(\displaystyle \int^5_1\sqrt{1+u^4}\,du\)?

56) Estimate \(\displaystyle \int^1_0t\,dt\) by comparison with the area of a single rectangle with height equal to the value of \(t\) at the midpoint \(t=\dfrac{1}{2}\). How does this midpoint estimate compare with the actual value \(\displaystyle \int^1_0t\,dt\)?

57) From the graph of \(\sin(2 \pi x)\) shown:

a. Explain why \(\displaystyle \int^1_0\sin(2 \pi t)\,dt=0\).

b. Explain why, in general, \(\displaystyle \int^{a+1}_a\sin(2 \pi t)\,dt=0\) for any value of \(a\).

Answer

a. The graph is antisymmetric with respect to \(t=\frac{1}{2}\) over \([0,1]\), so the average value is zero.
b. For any value of \(a\), the graph between \([a,a+1]\) is a shift of the graph over \([0,1]\), so the net areas above and below the axis do not change and the average remains zero.

58) If f is 1-periodic \((f(t+1)=f(t))\), odd, and integrable over \([0,1]\), is it always true that \(\displaystyle \int^1_0f(t)\,dt=0\)?

59) If f is 1-periodic and \(\displaystyle \int10f(t)\,dt=A\), is it necessarily true that \(\displaystyle \int^{1+a}_af(t)\,dt=A\) for all \(A\)?

Answer

Yes, the integral over any interval of length 1 is the same.