5.4: Indefinite Integrals and Net Change Theorem

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Throughout this module, if something does not exist, write DNE in the answer box.

Recap Video

Take a look at the following video which recaps the ideas from the section.

Example Video

Below is a video showing a worked example.

Problems

Evaluate the following:

$\displaystyle \int \sin (x)\,dx$ .
This is what type of integral? This means that we are looking for a general antiderivative of the function $f(x) = \sin (x)$ . Therefore, we can say $\int \sin (x)\,dx = \answer {-\cos (x)+C}.$
$\displaystyle \int _0^{\pi } \sin (x)\,dx$ .
This is what type of integral? This means that we are looking for a the signed area under the graph of $f(x) = \sin (x)$ on the interval $[0,\pi ]$ . To find the answer, we can use FTC part 2. One antiderivative of $f(x) = \sin (x)$ is $F(x) = \answer {-\cos (x)}+C$ . Therefore, $\int \sin (x)\,dx = -\cos (x)\Big |_0^{\pi } = \answer {2}.$

The indefinite integral $\displaystyle \int (\sec ^2(x)+\sec (x)\tan (x))\,dx = \answer {\tan (x)+\sec (x)+C}.$

Don’t forget the $+C$ .

$\displaystyle \int _0^1 (e^x+1)\,dx = \answer {e}$ .
$\displaystyle \int x^2(x+5)\,dx = \answer {\frac {1}{4}x^4+\frac {5}{3}x^3+C}$ .

Net Change Theorem Integrating the rate of change gives net change: $\int _a^b f'(x)\,dx = f(b)-f(a).$

A store produces bicycles at a rate of $b(t) = 100+3t^2-2t$ bikes/week. How many bikes are produced from the beginning of week $1$ to the end of week $2$ ?

We have the rate of production, and we want the actual production. To do this, we will integrate the rate over the time interval $[\answer {1},\answer {2}]$ . Therefore, it becomes $\int _1^2 b(t)\,dt = \answer {104}.$

The velocity of a car at time $t$ (in mph) is given by the following table: $\begin {array}{|c||c|c|c|c|c|} \hline $t$ & $0$ & $1/2$& $1$ & $3/2$ & $2$ \\ \hline $v(t)$ & $50$ & $60$ & $0$ & $-30$ & $-40$ \\ \hline \end {array}$

Use a left endpoint sum with $4$ subintervals to estimate the net change in position of the car on the interval $[0,2]$ . We get an approximation of $L_4 = \answer {40}$ milesmiles per hour .
Use a right endpoint sum with $4$ subintervals to estimate the net change in position of the car on the interval $[0,2]$ . We get an approximation of $R_4 = \answer {-5}$ milesmiles per hour .
Use a left endpoint sum with $4$ subintervals to estimate the total distance traveled by the car on the interval $[0,2]$ . We get an approximation of $L_4 = \answer {70}$ milesmiles per hour .
The total distance traveled means we need to integrate $|v(t)|$ instead of just $v(t)$ .
Use a right endpoint sum with $4$ subintervals to estimate the total distance traveled by the car on the interval $[0,2]$ . We get an approximation of $R_4 = \answer {65}$ milesmiles per hour .