2.1.2Is there an easier method?

Methods to Calculate Area Under a Curve

2-15.

Evaluate: $\displaystyle\sum _ { j = 2 } ^ { 20 } ( 2 j ^ { 2 } - 1 )$

2-16.

Discuss with your team how to use the summation feature of your graphing calculator to obtain the value of the summation below, where the function is $f(x) = 2\cos(\pi x)$ . Then evaluate the sum.

$\displaystyle\sum _ { i = 4 } ^ { 12 } 2 f ( 3 + 0.25 i )$

2-17.

This problem will help you develop a shortcut for writing and evaluating the summation of areas when approximating the area under a curve.

Given $f(x) = x^2 − 6x + 13$ , approximate the area under the curve for $2 \le x \le 5$ using six left endpoint rectangles as shown in the diagram below. Use the Estimating Area Under a Curve eTool (Desmos) to explore and verify your work. Click in the lower right corner of the graph to view it in full-screen mode.
Verify that the area of the first rectangle can be written as $A = \frac { 1 } { 2 }f(2)$ .

Then, complete the expanded sum below, which represents the total area under the curve.
Because all rectangles have a width of $\frac { 1 } { 2 }$ , we do not have to use an expanded sum to estimate the area. We can use sigma notation instead. Copy the sigma expression below and answer the following questions.
Use the summation feature of your graphing calculator to calculate the approximate area under the curve using your sigma expression from part (c). Compare your answer with the result from part (a). Try this using the Left Endpoint Rectangle eTool (Desmos). Click in the lower right corner of the graph to view it in full-screen mode.
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2-18.

Using the same function $f(x) = x^2 − 6x + 13$ , write an expression using sigma notation to approximate the area under the curve for $4 \le x \le 12$ using:

Two left endpoint rectangles of equal width.
Twenty-four left endpoint rectangles of equal width.
Nine left endpoint rectangles of equal width.
Explain why sigma notation requires that the rectangles have equal widths.

2-19.

Use the tables of data below to approximate the areas under the curves on the given domains. Use five left endpoint rectangles. Try this using the 2-19 Student eTool (Desmos). Click in the lower right corner of the graph to view it in full-screen mode.

$x$
$0$
$2$
$5$
$6$
$8$
$9$
$f(x)$
$4$
$5$
$10$
$11$
$12$
$14$
$x$
$0$
$2$
$4$
$6$
$8$
$10$
$f(x)$
$4$
$5$
$10$
$11$
$12$
$14$
Only one of the tables represents rectangles of equal widths. If you have not already done so, write an expression using sigma notation to estimate the area under the curve using five left endpoint rectangles.

$x$	$0$	$2$	$5$	$6$	$8$	$9$
$f(x)$	$4$	$5$	$10$	$11$	$12$	$14$

$x$	$0$	$2$	$4$	$6$	$8$	$10$
$f(x)$	$4$	$5$	$10$	$11$	$12$	$14$

2-20.

What is the difference, if any, between the values of $\displaystyle\sum _ { j = 3 } ^ { 8 } j ^ { 2 }$ and $\displaystyle\sum _ { j = 2 } ^ { 7 } ( j + 1 ) ^ { 2 }$ ? Homework Help ✎

2-21.

William wants to figure out how many calories he burns while at the gym. The number of calories he burns depends on which exercises he does during his workout. Given the graph below of the rate he burns calories, answer the questions below. Homework Help ✎

How many calories does William burn during the first hour of exercise? Support your answer with a short explanation of how you arrived at this result.
How many calories does William burn by the end of his workout?
What is the average number of calories that he burns per hour?

2-22.

Rewrite the summation notation $\displaystyle\sum _ { i = 6 } ^ { 11 } f ( i )$ so that the index goes from $i = 10$ to $i = 15$ and the result will be the same for any given function. Homework Help ✎

2-23.

Kristin is designing a model that will represent the path of a roller coaster. She has determined the beginning and the end parts of the track, but needs to write a formula for the middle section that will join the other segments. She decides that she wants one peak in this middle section, not including its boundaries. What values of $a$ and $b$ will make her model, given below, a continuous function? To help you visualize this, use the 2-23 HW eTool (Desmos). Homework Help ✎
$f ( x ) = \left\{ \begin{array} { l l } { - 2 \operatorname { cos } x + 3 } & { \text { for } x \leq 0 } \\ { a \operatorname { cos } ( b x ) - 2 } & { \text { for } 0 < x \leq 2 \pi } \\ { - \operatorname { cos } ( 2 x ) - 4 } & { \text { for } x > 2 \pi } \end{array} \right.$

2-24.

Given the graph of $y = h(x)$ at right, sketch: Homework Help ✎

$y = -h(x)$
$y = h(x) - 5$
$y = h^{-1}(x)$

Continuous piecewise on unscaled axis, left curve coming from upper left, opening up & turning at the highlighted point (negative 2, comma 7), right ray starting at, (negative 2, comma 7), continuing up & right

2-25.

Write a complete set of approach statements for $y = \frac { ( 3 x - 1 ) ( x + 2 ) } { 3 x - 1 }$ . Homework Help ✎

2-26.

If $f(x) = 2x + 3$ , calculate the area under the curve for $5 \le x \le 7$ . 2-26 HW eTool (Desmos). Homework Help ✎

2-27.

If the inverse of $f$ is a continuous function, why must the function be either strictly increasing or decreasing? Sketch an example to support your reasoning. Homework Help ✎

2-28.

Write an equation that will approximate the area under $f (x) = \cos(x)$ over the interval $[ - \frac { \pi } { 2 } , \frac { \pi } { 2 } ]$ using six trapezoids with equal height. How can you use the fact that this is an even function to save yourself some work? Homework Help ✎

Calculus Third Edition