4.1.1How can I calculate the exact area?

Definite Integrals

4-1.

THE RETURN OF FREDO AND FRIEDA

Examine these new velocity and distance graphs from Fredo and Frieda. Summarize how Fredo’s data is reflected in Frieda’s graph and how Frieda’s data is reflected in Fredo’s graph. You may want to review your results from Lesson 1.5.1.

Fredo’s Graph

First quadrant increasing curve, y axis labeled distance, opening up, starting at the origin, passing through the approximate points (120, comma 1400), & (240, comma 4200).

Frieda’s Graph

First quadrant continuous curve, y axis labeled velocity, with approximate turning points as follows: starting @ the origin, turning right at (35, comma 14), turning up @ (115, comma 15), turning right @ (120, comma 24), turning up @ (200, comma 25), turning right @ (215, comma 32), ending @ (250, comma 32).

Since each student’s data is confirmed by the other student’s data, using a derivative (to determine the slope of Fredo’s graph) must be linked to using an integral (to calculate the area under Frieda’s graph). Explain the forward and backward nature of the connection between slope and area.
Recall that the slope of a secant line can be used to determine the average velocity (approximate slope) at any point on Fredo’s curve. How can the exact slope of a curve at a point be found?
Rectangles can be used to approximate the area under Frieda’s curve in order to calculate the distance. Make a conjecture about how the exact area under a curve can be determined.

4-2.

Regardless of what value of $n$ is chosen, a Riemann sum can only approximate the area under $f$ on $[a, b]$ because the rectangles either add extra area or miss some area. Some values of $n$ give better approximations than others.

First quadrant labeled 4 rectangles, curve labeled, f of x, coming from lower left, turning down, then turning up, & 4 vertical shaded bars, of equal widths, bottom edges on x axis, left edge of first bar labeled, a, right edge of last bar labeled, b, with top left vertex of each bar, on the curve. First quadrant labeled 8 rectangles, curve labeled, f of x, coming from lower left, turning down, then turning up, & 8 vertical shaded bars, of equal widths, bottom edges on x axis, left edge of first bar labeled, a, right edge of last bar labeled, b, with top left vertex of each bar, on the curve. First quadrant labeled 16 rectangles, curve labeled, f of x, coming from lower left, turning down, then turning up, & 16 vertical shaded bars, of equal widths, bottom edges on x axis, left edge of first bar labeled, a, right edge of last bar labeled, b, with top left vertex of each bar, on the curve.

Examine the graphs above and write down your observations.
How can we calculate an exact area? Using complete sentences, describe your ideas thoroughly.
The width of each rectangle is $\frac { b - a } { n }$ and represented by $Δx$ . What happens to $Δx$ as more rectangles are used?

4-3.

CALCULATING EXACT AREA

Use a Riemann sum to write an expression to represent the exact area under $f$ on $[a, b]$ .
Will $Δx$ ever equal $0$ ? Why or why not?
What happens to the area of each individual rectangle as $n → ∞$ ?
If the area is composed of rectangles with areas that are approaching zero, why does the overall area not approach zero?

Math Notes

Definite Integrals

A Riemann sum is a convenient way to approximate the area under a curve using rectangles. However, in order to get an exact area, an infinite number of rectangles is needed! Since we cannot algebraically evaluate the sum of infinitely many rectangles, we use a limit, the limit as the number of rectangles approaches infinity. Another way to express a limit such as this is with a definite integral, shown below:

$\lim\limits _ { n \rightarrow \infty } \displaystyle\sum _ { i = 0 } ^ { n - 1 } ( \frac { b - a } { n } ) f ( a + ( \frac { b - a } { n } ) i ) = \int _ { a } ^ { b } f ( x ) d x$

This “limit of a Riemann sum,” shown above represents the exact area under the graph of $y = f\left(x\right)$ on a closed interval $[a, b]$ . Each rectangle will have an infinitesimally small width expressed as $\lim\limits _ { n \rightarrow \infty } \frac { b - a } { n }$ or $dx$ . The symbol $\int$ represents the “ $S$ ” of “summe,” the German word for sum.

Note that all rectangles in this definition are of equal width. It is possible to define the definite integral when the widths of the intervals are different, as long as all of the widths approach zero.

For example, the exact area under the curve $f(x) = x^{2} + 1$ over the interval $–2 ≤ x ≤ 3$ is represented by $\lim\limits _ { n \rightarrow \infty } \displaystyle\sum _ { i = 0 } ^ { n - 1 } ( \frac { 3 - ( - 2 ) } { n } ) f ( - 2 + ( \frac { 3 - ( - 2 ) } { n } ) i ) = \int _ { a } ^ { b } f ( x ) d x$

In the definite integral above, $–2$ is the lower bound, $3$ is the upper bound, and the expression $x^{2} + 1$ is called the integrand.

Definite integrals can also be used on functions with removable or jump discontinuities.

4-4.

Examine the general form of a definite integral, $\int _ { a } ^ { b } f ( x ) d x$ ,as shown in the preceding Math Notes box.

What do the upper “ $b$ ,” and lower “ $a$ ,” bounds of the definite integral represent?
The Math Notes box states that $\int _ { a } ^ { b } f ( x ) d x$ is equivalent to $\lim\limits_ { n \rightarrow \infty } \displaystyle\sum _ { i = 0 } ^ { n - 1 } ( \frac { b - a } { n } ) f ( a + ( \frac { b - a } { n } ) i )$ . Compare $\frac { b - a } { n }$ in the limit to $dx$ in the definite integral.
Explain why it is important to remember that we are multiplying $\left(x^{2} + 1\right)$ by $dx$ .

4-5.

For the following definite integrals, sketch each function and shade the appropriate region. Describe the region and then calculate the area without using a calculator.

$\int _ { 0 } ^ { 2 \pi } \operatorname { sin } ( 2 t ) d t$
$\int _ { - 2 } ^ { 3 } ( \frac { 1 } { 2 } x + 3 ) d x$
$\int _ { 0 } ^ { 6 } 4 d x$
$\int _ { 4 } ^ { 4 } ( 3 k ^ { 2 } ) d k$

Compute without a calculator

4-6.

While driving to work, Mr. Matluck’s velocity is given by $v\left(t\right) = 15t + 10$ , where $t$ is measure in hours and $v\left(t\right)$ is miles per hour. Determine how far Mr. Matluck lives from school if it takes him: Homework Help ✎

$1$ hour to get to work.
$4$ hours to get to work.
$\frac { 1 } { 2 }$ hour to get to work.
$t$ hours to get to work.

First quadrant increasing line, from (0, comma 10), to (4, comma 70).

4-7.

Examine the function, $f$ , graphed at right. Homework Help ✎

Is $f$ even, odd, or neither?
If $\int _ { 0 } ^ { 2 } f ( x ) d x = 10$ , what is $\int _ { - 2 } ^ { 2 } f ( x ) d x$ ? Explain.
If $\int _ { 0 } ^ { 3 } f ( x ) d x = - 2$ and $\int _ { 0 } ^ { 2 } f ( x ) d x = 10$ , what is the value of $\int _ { 2 } ^ { 3 } f ( x ) d x$ ? Explain.
If you know the values of $\int _ { 0 } ^ { 3 } f ( x ) d x$ and $\int _ { - 2 } ^ { 2 } f ( x ) d x$ , how can you determine the value of $\int _ { 2 } ^ { 3 } f ( x ) d x$ ? Justify your process with a diagram, if necessary.

4-8.

For each function, write the equation of its slope function, $f ^\prime$ . Homework Help ✎

$f\left(x\right)=\frac{2}{5}x^{-2}-4x$
$f(x)=-2\sqrt{x}$
$f\left(x\right)=6\cos\left(x\right)$
$f ( x ) = \frac { 4 x ^ { 2 } + 4 } { x ^ { 2 } + 1 }$

4-9.

Write a Riemann sum that approximates the area under the curve for $–8 ≤ x ≤ 8$ using the given number of rectangles if $f\left(x\right) = 2x^{1/3} + 1$ . Then evaluate the sum with a calculator. Homework Help ✎

$16$ rectangles
$64$ rectangles

4-10.

Use the Power Rule to write an equation for $f^\prime$ if $f\left(x\right) = \left(x + 4\right)\left(x – 3\right)$ by first expanding $f$ . Then, write the equation of the line tangent to $f$ at $x = 3$ . Homework Help ✎

4-11.

Write a Riemann sum that approximates the area under the curve for $2 ≤ x ≤ 5$ using $n$ rectangles given $f\left(x\right) = 3x + 5$ . Then approximate the area using at least four different values of $n$ . For what value of $n$ is your approximation most accurate and why? Homework Help ✎

4-12.

Khi thinks that when the flags at right are rotated, the $\left(\text{volume of }A\right)+$ $\left(\text{volume of }C\right)=$ $\left(\text{volume of}\left(B+C\right)\right)$ .

Decide if Khi is correct. If he is incorrect, explain the error in his logic. 4-12 HW eTool Homework Help ✎

Horizontal segment, with shaded right triangle labeled, A, above segment, with its short leg, labeled 3, on the right third of the segment, & vertical leg labeled, 4.

Horizontal segment, with horizontal rectangle labeled, C, above segment, with its bottom edge, labeled 3, on the right third of the segment, & right edge labeled 2, with shaded right triangle labeled, B, sharing its horizontal leg with top edge of rectangle, vertical leg aligned with right edge of rectangle, labeled 4.

Calculus Third Edition

Definite Integrals