2.1.3Who was Riemann?

Area Under a Curve as a Riemann Sum

2-29.

We will examine the graph from problem 2-17 again, but this time we will use twelve left endpoint rectangles to approximate the area under the curve for $2\le x\le5 \text{ if } f(x)=x^2-6x+13$ . Use the Estimating Area Under a Curve eTool to explore and verify your work. Click in the lower right corner of the graph to view it in full-screen mode.

Write an expanded sum using the area of each rectangle.
Write and evaluate the summation using sigma notation.
Compare this result to those from problem 2-17. Which is more accurate and why?

Math Notes

Approximating Area Using Left Endpoint Rectangles

The area under $f$ from $x=a$ to $x=b$ can be approximated by summing the areas of $n$ left endpoint rectangles. In the diagram at right, the shaded rectangle is a typical rectangle, one that represents all rectangles across the region.

If each rectangle has a width of $\Delta x$ , then the summation can be written in sigma notation as follows:

$A \approx \displaystyle\sum _ { i = 0 } ^ { n - 1 } [ \Delta x \cdot f ( a + \Delta x \cdot i ) ]$

Using rectangles to approximate area under the curve is generally known as a Riemann sum, named in honor of George Friedrich Bernhard Riemann (1826 – 1866).

First quadrant, x axis with 2 tick marks, first about 1 fourth right labeled, a, second about 3 fourths right labeled, b, increasing curve labeled f of x, opening up coming through the y axis, about 1 sixth up, continuing up & right, & 1 vertical shaded bar, bottom edge, labeled delta x, on x axis, center edof x axis, with top left edge vertex on the curve.

2-30.

Use summation notation to write an expression that will approximate the area under the curve for a function $f$ over each interval below, using the specified number of left endpoint rectangles.

$3 \le x \le 10; 21$ rectangles

$−2 \le x \le 6; 10$ rectangles

2-31.

Rewrite your summation expression from part (a) of problem 2-30 so that right endpoint rectangles are used to approximate the area instead.

2-32.

Write a general expression using summation notation that can be used to approximate area under a curve using right endpoint rectangles.

2-33.

Xavier likes things to be exact—he has grown weary of overestimates and underestimates. He thinks he has found a way to calculate the exact area under a curve: midpoint rectangles! Is Xavier correct? Justify your answer by sketching different functions and shading the midpoint rectangles.

2-34.

The estimation of the area under the curve for $2 \le x \le 6$ where $f(x)=\sqrt{x-2}$ is shown at right using four midpoint rectangles.

Use sigma notation to write a Riemann sum that describes the given situation.
If the rectangles used in part (a) are rotated about the $x$ -axis, we could use the resulting figure to estimate the volume of the rotated flag. Describe the resulting three-dimensional shape. Include a sketch.
Estimate the volume of this rotated region by calculating the volume of each rotated rectangle. How reasonable is this result?

First quadrant grid on integer values from 0 to 7 on x axis, & from 0 to 4 on y axis, increasing curve opening down, starting at the point (2, comma 0), passing through (6, comma 2), & 4 shaded vertical bars, each with width of 1, starting at x = 2, bottom edges of bars on the x axis, midpoint of top edge of each bar, on the curve.

2-35.

Construct a composite function $k(x)$ using $f(x) =\sqrt { x }$ and $g(x) = \log(x)$ if $k(x) =\frac { 1 } { 2 }\log(x)$ . Homework Help ✎

2-36.

Write a thorough description of the function $y = \frac { x ^ { 2 } + 2 x - 15 } { x - 2 }$ . Include a slope statement, a complete set of approach statements, and describe its end behavior. 2-36 HW eTool. Homework Help ✎.

2-37.

Calculate the value of the given summation and explain how you found your answer. Homework Help ✎

$\displaystyle\sum _ { p = 1 } ^ { 80 } \operatorname { cos } ( \frac { \pi p } { 2 } )$

2-38.

Is the inverse of an odd function also a function? If the inverse is a function, is it also an odd function? How do you know? Include a statement to support your answer and sketch a graph of an example. Investigate this using the Draw Inverse eToo. Homework Help ✎.

2-39.

Calculate the volume of the solid formed by rotating the flag bound by the $x$ - and $y$ -axes and $y = \sqrt{9 - x^2}$ , in the first quadrant, about the $y$ -axis. Homework Help ✎

2-40.

For the given function, write an expression using summation notation that will approximate the area under the curve for $3 \le x \le 7$ using eight rectangles. Specify if you use left, right, or midpoint rectangles. Then enter this summation expression into your graphing calculator and evaluate the approximate area. Homework Help ✎

$f(x) = \frac { 2 ( x + 4 ) } { x + 6 }$

2-41.

Rewrite each of the following sums using summation notation. Homework Help ✎

$5 + 7 + 9 + 11 + 13$
$2\cos(2\pi) + 3\cos(3\pi) + 4\cos(4\pi) + 5\cos(5\pi)$
$\frac { 1 } { 5 } f ( - 2 ) + \frac { 1 } { 5 } f ( - 1 ) + \frac { 1 } { 5 } f ( 0 ) + \frac { 1 } { 5 } f ( 1 ) + \frac { 1 } { 5 } f ( 2 )$

Calculus Third Edition

Approximating Area Using Left Endpoint Rectangles