Question

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

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

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

Question

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

Question

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

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 \(A=4.\)

Question

Question

Question

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

Question

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

Question

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

Question

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

Question

Question

Question

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

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

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

Question

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

Question

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

Question

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

Question

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

Question

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

Question

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

Question

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

Question

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

Question

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

Question

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

Question

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

Question

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

Question

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

Question

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

Question

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

Question

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

Question

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

Question

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

Question

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

Question

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

Question

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

Question

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

Question

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

Question

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

Question

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

Question

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

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

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

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

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 \(B\).