Question 5.2

Explain how rectangles can be used to approximate the areaenclosed by the graph of a function \(y=f(x) \geq 0,\) the\(x\)-axis, and the lines \(x=a\) and \(x=b.\)

Question 5.3

True or False  When a closed interval \([a,b] \) is partitioned into \(n\) subintervals each of the same length, the lengthof each subinterval is \(\dfrac{a+b}{n}\).

Question 5.4

If the closed interval \([-2,4]\) is partitionedinto \(12\) subintervals, each of the same length, then the length of each subinterval is ______.

Question 5.5

True or False  If the area \(A\) under the graph of afunction \(f\) that is continuous and nonnegative on a closedinterval \([a,b] \) is approximated using upper sums \(S_{n}\), then \(S_{n}\geq A\)and \(A=\lim\limits_{n\rightarrow \infty }S_{n}.\)

Question 5.6

Approximate the area \(A\) enclosed bythe graph of \(f(x) = \dfrac{1}{2}x+3,\) the \(x\)-axis, and the lines \(x=2\) and \(x=4\) by partitioning the closed interval \([2, 4] \) into four subintervals:\[\left[ 2, \dfrac{5}{2}\right], \left[ \dfrac{5}{2},3\right], \left[3, \dfrac{7}{2}\right], \left[ \dfrac{7}{2}, 4\right].\]

1. Using the left endpoint of each subinterval, draw four small rectanglesthat lie below the graph of \(f\) and sum the areas of the four rectangles.
2. Using the right endpoint of each subinterval, draw four smallrectangles that lie above the graph of \(f\) and sum the areas of the fourrectangles.
3. Compare the answers from parts (a) and (b) to the exactarea \(A=9 \) and to the estimates obtained in Example 1.

Question 5.7

Approximate the area \(A\) enclosed by the graph of \(f(x) =6-2x,\)the \(x\)-axis, and the lines \(x=1\) and \(x=3\) by partitioning the closed interval \([ 1,3] \) into four subintervals:\[\left[1, \dfrac{3}{2}\right], \left[ \dfrac{3}{2},2\right], \left[ 2,\dfrac{5}{2}\right], \left[ \dfrac{5}{2},3\right].\]

1. Using the right endpoint of each subinterval, draw four smallrectangles that lie below the graph of \(f\) and sum the areas of the fourrectangles.
2. Using the left endpoint of each subinterval, draw four small rectanglesthat lie above the graph of \(f\) and sum the areas of the four rectangles.
3. Compare the answers from parts (a) and (b) to the exactarea \(A=4.\)

Question 5.8

Question 5.9

Question 5.10

\([1,4]\) with \(n = 3\)

Question 5.11

\([0,9]\) with \(n = 9 \)

Question 5.12

\([{-}1,4]\) with \(n = 10 \)

Question 5.13

\([{-}4,4]\) with \(n = 16\)

Question 5.14

Question 5.15

Question 5.16

Area Under a Graph Consider the area under the graph of \(y=x\) from \(0\) to \(3\).

1. Sketch the graph and the area under the graph.
2. Partition the interval \([0,3]\) into \(n\) subintervals each of equal length.
3. Show that \(s_{n}=\sum\limits_{i=1}^{n}(i-1)\left(\dfrac{3}{n}\right)^{2}\).
4. Show that \(S_{n}=\sum\limits_{i=1}^{n}{i\left(\dfrac{3}{n}\right)}^{2}\).
5. Show that \(\lim\limits_{n\rightarrow \infty}s_{n}=\lim\limits_{n\rightarrow \infty} S_{n}={\dfrac{{9}}{{2}}}\).

Question 5.17

Area Under a Graph Consider the area under the graph of \(y=4x\) from \(0\) to \(5\).

1. Sketch the graph and the corresponding area.
2. Partition the interval \([0,5]\) into \(n\) subintervals each of equal length.
3. Show that \(s_{n}=\sum\limits_{i=1}^{n}(i-1){\dfrac{100}{n^{2}}}\).
4. Show that \(S_{n}=\sum\limits_{i=1}^{n} i{\dfrac{100}{n^{2}}}\).
5. Show that \(\lim\limits_{n\rightarrow \infty}s_{n}={\lim\limits_{n\rightarrow \infty }}S_{n}=50.\)

Question 5.18

\(f(x) = -x+10\) on \([0,8]\)

Question 5.19

\(f(x) = 2x + 5\) on \([2, 6]\)

Question 5.20

\(f(x) = 16-x^{2}\) on \([0,4]\)

Question 5.21

\(f(x) = x^{3}\) on \([0,8]\)

Question 5.22

\(f(x) = \cos x\) on \(\left[ -\dfrac{\pi}{2},\dfrac{\pi }{2}\right] \)

Question 5.23

\(f(x) = \sin x\) on\([0,\pi]\)

Question 5.24

Rework Example 3 by using lower sums \(s_{n}\) (rectangles that liebelow the graph of \(f\)).

Question 5.25

Rework Example 4 by using upper sums\(S_{n}\) (rectangles that lie above the graph of \(f\)).

Question 5.26

\(f(x) =2x+1\) from \(a = 0\) to \(b = 4 \)

Question 5.27

\(f(x) = 3x+1\) from \(a=0\) to \(b=4\)

Question 5.28

\(f(x) =12-3x\) from \(a = 0\) to \(b = 4\)

Question 5.29

\(f(x) =5-x\) from \(a = 0\) to \(b=4\)

Question 5.30

\(f(x) = 4 x^{2}\) from \(a = 0\) to \(b = 2\)

Question 5.31

\(f(x) = \dfrac{1}{2}x^{2}\) from \(a = 0\) to \(b=3\)

Question 5.32

\(f(x) = 4 - x^{2}\) from \(a = 0\) to \(b = 2\)

Question 5.33

\(f(x) = 12-x^{2}\) from \(a = 0\) to \(b = 3\)

Question 5.34

\(f(x)=x+3\) from \(a = 1\) to \(b=3\)

Question 5.35

\(f(x)=3-x\) from \(a=1\) to \(b=3\)

Question 5.36

\(f(x)=2x+5\) from \(a=-1\) to \(b=2\)

Question 5.37

\(f(x)=2-3x\) from \(a=-2\) to \(b=0\)

Question 5.38

\(f(x) = 2x^{2}+1\) from \(a = 1\) to \(b = 3\)

Question 5.39

\(f(x) = 4-x^{2}\) from \(a = 1\) to \(b = 2\)

Question 5.40

\(f(x)=xe^{x}\) on \([0,8]\)

Question 5.41

\(f(x)=\ln x\) on \([1,3] \)

Question 5.42

\(f(x)=\dfrac{1}{x}\) on \([1,5]\)

Question 5.43

\(f(x)=\dfrac{1}{x^{2}}\) on \([2,6]\)

Question 5.44

1. Graph \(y=\dfrac{4}{x}\) from \(x=1\) to \(x=4\) and shade the area under its graph.
2. Partition the interval \([1,4] \) into \(n\) subintervals of equal length.
3. Show that the lower sum \(s_{n}\) is \(s_{n}=\sum\limits_{i=1}^{n}\dfrac{4}{\left( 1+\dfrac{3i}{n}\right) }\left(\dfrac{3}{n}\right).\)
4. Show that the upper sum \(S_{n}\) is \[S_{n}=\sum\limits_{i=1}^{n}\frac{4}{\left( 1+\frac{3(i-1) }{n}\right) }\left( \frac{3}{n}\right)\]
5. Complete the following table:

   \(n\)	\(5\)	\(10\)	\(50\)	\(100\)
   \(s_{n}\)
   \(S_{n}\)

6. Use the table to give an upper and lower bound for the area.

Question 5.45

Area Under a Graph  Approximate the areaunder the graph of \(f(x)=x\) from \(a\geq 0\) to \(b\) by using lower sums \(s_{n}\) andupper sums \(S_{n}\) for a partition of \([a,b]\) into \(n\)subintervals, each of length \(\dfrac{b-a}{n}\). Show that\[s_{n}\lt \frac{b^{2}-a^{2}}{2}\lt S_{n}\]

Question 5.46

Area Under a Graph  Approximate the areaunder the graph of \(f(x)=x^{2}\) from \(a\geq 0\) to \(b\) byusing lower sums \(s_{n}\) and upper sums \(S_{n}\) for a partition of \([a,b]\) into \(n\)subintervals, each of length \(\dfrac{b-a}{n}\). Show that \[s_{n}\lt \frac{b^{3}-a^{3}}{3}\lt S_{n}\]

Question 5.47

Area of a RightTriangle  Use lower sums \(s_{n}\) (rectangles thatlie inside the triangle) and upper sums \(S_{n}\) (rectangles thatlie outside the triangle) to find the area of a right triangle ofheight \(H\) and base \(B.\)

Question 5.48

Area of a Trapezoid  Use lower sums \(s_{n}\) (rectangles that lie inside the trapezoid) and upper sums \(S_{n}\) (rectangles that lie outside the trapezoid) to find the area of a trapezoid of heights \(H_1\) and \(H_2\) and base \(B\).