5.1 Assess Your Understanding

Concepts and Vocabulary

Question 1.

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

Question 2.

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

Question 3.

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

Question 4.

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

351

Skill Building

Question 5.

Approximate the area \(A\) enclosed by the 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 rectangles that lie below the graph of \(f\) and sum the areas of the four rectangles.
  2. Using the right endpoint of each subinterval, draw four small rectangles that 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 exact area \(A=9 \) and to the estimates obtained in Example 1.

Question 6.

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 small rectangles that lie below the graph of \(f\) and sum the areas of the four rectangles.
  2. Using the left endpoint of each subinterval, draw four small rectangles that 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 exact area \(A=4.\)

In Problems 7 and 8, refer to the graphs below. Approximate the shaded area under the graph of \(f\):

  1. By constructing rectangles using the left endpoint of each subinterval.
  2. By constructing rectangles using the right endpoint of each subinterval.

Question 7.

Question 8.

In Problems 9–12, partition each interval into \(n\) subintervals each of the same length.

Question 9.

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

Question 10.

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

Question 11.

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

Question 12.

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

In Problems 13 and 14, refer to the graphs. Approximate the shaded area:

  1. By using lower sums \(s_{n}\) (rectangles that lie below the graph of \(f\)).
  2. By using upper sums \(S_{n}\) (rectangles that lie above the graph of \(f\)).

Question 13.

Question 14.

Question 15.

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 16.

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 \(\lim\limits_{n\rightarrow \infty} s_{n}={\lim\limits_{n\rightarrow \infty }}S_{n}=50.\)

In Problems 17–22, approximate the area \(A\) under the graph of each function \(f\) from \(a\) to \(b\) for \(n = 4\) and \(n = 8\) subintervals:

  1. By using lower sums \(s_{n}\) (rectangles that lie below the graph of \(f\)).
  2. By using upper sums \(S_{n}\) (rectangles that lie above the graph of \(f\)).

Question 17.

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

Question 18.

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

Question 19.

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

Question 20.

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

352

Question 21.

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

Question 22.

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

Question 23.

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

Question 24.

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

In Problems 25–32, find the area \(A\) under the graph of \(f\) from \(a\) to \(b\):

  1. By using lower sums \(s_{n}\) (rectangles that lie below the graph of \(f\)).
  2. By using upper sums \(S_{n}\)(rectangles that lie above the graph of \(f\)).
  3. Compare the work required in (a) and (b). Which is easier? Could you have predicted this?

Question 25.

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

Question 26.

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

Question 27.

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

Question 28.

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

Question 29.

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

Question 30.

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

Question 31.

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

Question 32.

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

Applications and Extensions

In Problems 33–38, find the area under the graph of \(f\) from \(a\) to \(b.\) [Hint: Partition the closed interval \([a,b]\) into \(n\) subintervals \([x_{0},x_{1}],[x_{1},x_{2}],\) \(\ldots ,\) \([x_{i-1},x_{i}],\) \(\ldots , [x_{n-1},x_{n}],\) where \(a=x_{0}\lt x_{1}\lt\cdots \lt x_{i}\lt\cdots \lt x_{n-1}\lt x_{n}=b,\) and each subinterval is of length \(\Delta x=\dfrac{b-a}{n}.\) As the figure below illustrates, the endpoints of each subinterval, written in terms of \(n\), are \[ \begin{eqnarray*} x_{0} &=& a, x_{1}=a+\dfrac{b-a}{n},\enspace x_{2}=a+2\left(\dfrac{b-a}{n}\right), \ldots , \\ x_{i-1} &=& a+(i-1) \left(\dfrac{b-a}{n} \right),\enspace x_{i}=a+i\left(\dfrac{b-a}{n}\right), \ldots ,\\ x_{n} &=& a+n \left(\dfrac{b-a}{n}\right) \end{eqnarray*} \]

Question 33.

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

Question 34.

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

Question 35.

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

Question 36.

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

Question 37.

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

Question 38.

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

In Problems 39–42, approximate the area \(A\) under the graph of each function \(f\) by partitioning \([a,b]\) into \(20\) subintervals of equal length and using an upper sum.

Question 39.

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

Question 40.

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

Question 41.

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

Question 42.

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

Question 43.

  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.

Challenge Problems

Question 44.

Area Under a Graph  Approximate the area under the graph of \(f(x)=x\) from \(a\geq 0\) to \(b\) by using 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 \[ s_{n}\lt \frac{b^{2}-a^{2}}{2}\lt S_{n} \]

Question 45.

Area Under a Graph  Approximate the area under the graph of \(f(x)=x^{2}\) from \(a\geq 0\) to \(b\) by using 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 46.

Area of a Right Triangle  Use lower sums \(s_{n}\) (rectangles that lie inside the triangle) and upper sums \(S_{n}\) (rectangles that lie outside the triangle) to find the area of a right triangle of height \(H\) and base \(B.\)

Question 47.

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 \(B\).

353