4.2.5What is an integral, really?

A car is traveling along a straight road with a velocity given by the function $v$ such that $v\left(t\right)$ represents miles per hour. Examine the equation  $\int _ { 3 } ^ { 7 } v ( t ) d t = 120$.

1. According to the context of the problem, what does the value $120$ represent? What are its units? Explain how you know.

2. Anahit claimed that $120$ represents the “area under $v\left(t\right)$” and the units should be “square units.” Discuss her answer with your team. Is she correct?

While area under the curve offers a convenient way to think about integrals, integrals can be used in a variety of contexts.

1. For each situation below, write a complete description of what the integral is computing. Use correct units and be sure to mention the meaning of the bounds in your description.

   1. $\int _ { 0 } ^ { 75 } P ( c ) d c = 125.93$ where $P$ represents the rate that a coffee shop makes a profit in dollars per ounces of coffee sold, $c$.

   2. $\int _ { 150 } ^ { 2000 } H ( r ) d r = 13$ where $H$ represents the rate that a bee produces honey in teaspoons per round-trips, $r$, that a bee takes to a flower. 

   3. $\int _ { 3 } ^ { 7 } | v ( t ) | d t = 360$ where $v\left(t\right)$ represents the velocity of a car traveling on a north-south highway in miles per hour and $t\left(0\right) = 8$ a.m.

2. What do all of the definite integrals in part (a) have in common? What does the integration operation do to the integrand? Discuss these questions with your team and be prepared to share your answers with the class.

3. Antiderivatives are often called accumulation functions because they compute net change of the integrand over a given interval. For example, an antiderivative of $v$, where $v\left(t\right)$ represents velocity as function of time (a.k.a. distance/time), will compute displacement (net change in change in distance) over a given interval. Therefore, as an operation, definite integrals can be interpreted as accumulators. With this interpretation in mind, describe what accumulates in each of the examples in part (a). 

Luis’s hair started going gray when he was a young child. On the morning of his $17$^th birthday (which is the day of a big Calculus test), he counts $70$ strands of white hair. His mother, who keeps data on her precocious child, notices that the rate $R\left(t\right)$ that white hair appears on Luis’s head can be modeled by the function $R\left(t\right) = e^{0.14t}$, where $t$ is time in years.

1. Examine the graph of $y = R\left(t\right)$ on your calculator. What does the area under the curve on the interval $[17, 25]$ represent? In other words what is accumulated? 

2. Does the amount of gray hair on Luis’s head increase or decrease as he gets older? Use the graph of $y = R\left(t\right)$ to justify your answer.

3. Write and evaluate an integral expression to represent the number of white hairs that Luis had on the day he was born. Then, write and evaluate an integral to represent the number of white hairs that will have accumulated on his $21$^st birthday.

4. Luis did some research and discovered that an average human head has $10,000$ hairs on their head. Assuming that Luis will not go bald, use your calculator to determine what decade of his life he should expect to have all white hair? 

You have entered a lollipop eating competition. Each contestant is given a large spherical lollipop with a volume of $2400$ cm³. The first contestant to lick their way to the center of the lollipop is the winner. You have been practicing for months and have found that as you lick away towards the center, the rate that the radius of the lollipop changes (in centimeters per minute) can be modeled by the function $R ( t ) = ( - \frac { 3 } { 2500 } t ) \operatorname { ln } ( 9 t )$.

1. Use your calculator to examine the graph of $y = R\left(t\right)$ during the first hour of the competition. Is $R\left(t\right)$ positive or negative? Is $R\left(t\right)$ increasing or decreasing? Interpret the graph in the context of the problem. 

2. Interpret the expression $\int _ { 0 } ^ { 10 } R ( t ) d t$ in the context of this problem. What accumulates? Then use your calculator to evaluate this expression. Use correct units in your answer.

3. According to the function $R\left(t\right)$, how much of the lollipop (in cubic centimeters) will remain after $10$ minutes? 

4. Last year, the winner took exactly $50$ minutes to finish a $2400$ cm³ lollipop. Do you have a good chance of beating last year’s record? 

Daniel the Dinosaur sees a meteor heading directly towards Earth and fears doom. He knows that the mass of the meteor (in kilograms) will decrease as it enters Earth’s atmosphere, and the rate the mass decreases can be modeled by the function $K ( m ) = - \frac { 1 } { 20 } e ^ { m / 20 }$, where $m$ is measured in kilometers and $K\left(m\right)$ is measured in kilograms/kilometer. At the moment Daniel spots the meteor, it is $100$ kilometers from Earth and has a mass of $199$ kg. If the mass is greater than $50$ kg on impact, then Daniel and all dinosaurs will be destroyed. Set up and evaluate an integral expression that will determine if Daniel and the dinosaurs will survive.

Brianna is babysitting for her calculus teacher because she broke her phone and needs to buy a new one. She will get paid at a flat rate of $$3.50$ per hour, per child. Natalie is away at a friend’s house for the first two hours while Brianna is babysitting Morgan and Lydia. Natalie returns and Brianna continues to babysit for three additional hours.

1. Sketch a graph and write a piecewise-defined function to represent Brianna’s pay rate.

2. How much will Brianna get paid for the five hours of work?

3. Represent Brianna’s total pay using definite integrals. 

$\int _ { 1 } ^ { 4 } S ( t ) d t = 12$, where $S$ represents the rate that snow falls in a driveway in cubic inches per hour and $t$ is hours after $5$ p.m.

Write a complete description about what the value $12$ represents. In other words, in the context of the problem, what is accumulated? Use correct units and be sure to mention the meaning of the bounds in your description. Homework Help ✎

Evaluate each integral below. What is the difference between them? Homework Help ✎

1. What is $\int _ { 0 } ^ { x } ( 2 t + 5 ) d t$?

2. What is $\int _ { 4 } ^ { 9 } ( 2 x + 5 ) d x$?

In Lesson 4.2.3 you should have noticed a close relationship between derivatives and integrals just like with velocity and distance. In particular, you have seen that if you know the velocity, $v\left(t\right)$, at time $t$ then you can compute the distance traveled from $t = 0$ to $t = x$ by using $s ( x ) = \int _ { 0 } ^ { x } v ( t ) d t$. Homework Help ✎

1. If $v\left(t\right) = 2t + 5$ in miles per hour, how far has the car traveled after $2$ hours? After $5$ hours? After $x$ hours?

2. How far did the car travel between $2$ and $4$ hours?

3. In Chapter 3 you considered the instantaneous rate of change (derivative) of any function. Explain why you expect the derivative, $s^\prime\left(t\right)$, of a distance function of a car to be the velocity function, $v\left(t\right)$, of the car.

Determine if each of the following conjectures is always true, sometimes true, or never true. Be sure to provide examples and/or counterexamples to support your claim. Homework Help ✎

Conjecture 1: If the slope of the first derivative of a function is negative over the interval $[a, b],$ then a local maximum of that function will exist somewhere in that interval.

Conjecture 2: Local extrema of a function are found where the derivative equals zero.

Conjecture 3: Local minima exist at the $x$-values where the first derivative changes from positive to negative.

Conjecture 4: When the first derivative and the second derivative both equal zero, then the function has both a local extrema and an inflection point.

A function $f$ is a continuous, odd, and$\lim\limits _ { x \rightarrow - \infty } f ( x ) = 5$. Sketch a graph of $y = f\left(x\right)$. Homework Help ✎

For $f\left(x\right)=\sin\left(x^2\right)$, $g ( x ) = \sqrt { x - 2 }$, and $h(x)=\frac{1}{x}$, write equations for each of the following compositions of functions and state the domains. Homework Help ✎

1. $f\left(g\left(x\right)\right)$

2. $f\left(g\left(h\left(x\right)\right)\right)$

3. $h\left(f\left(g\left(x\right)\right)\right)$

4. $h\left(h\left(x\right)\right)$

For $y = x^{3} + 3x^{2} – 24x$, over what interval(s) is $y^\prime$ increasing? Homework Help ✎

For each function below, write a Riemann sum using $16$ rectangles to estimate the area under the curve over the interval $–4 ≤ x ≤ 4$. Then use your calculator to evaluate the sum. Recall that a Riemann sum does not have to involve sigma notation. Homework Help ✎

1. $g\left(x\right)=x\sin^2\left(x\right)$

2. $f ( x ) = \left\{ \begin{array} { c } { 3 \text { for } x < 0 } \\ { x ^ { 2 } + 3 \text { for } x \geq 0 } \end{array} \right.$