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Calculus for Biology and Medicine

Claudia Neuhauser

Chapter 6

Integration - all with Video Answers

Educators

Section 1

The Definite Integral

05:32

Problem 1

Approximate the area under the parabola $y=x^{2}$ from 0 to 1 , using four equal subintervals.

Willis James
Willis James
Numerade Educator
05:32

Problem 2

Approximate the area under the parabola $y=x^{2}$ from 0 to 1 . using five equal subintervals.

Willis James
Willis James
Numerade Educator
02:06

Problem 3

Approximate the area under the curve $y=x^{3}$ from 0 to 1 , using six equal subintervals.

Vinnu M
Vinnu M
Numerade Educator
05:32

Problem 4

Approximate the area under the parabola $y=1-x^{2}$ from 0
to 1, using five equal subintervals.

Willis James
Willis James
Numerade Educator
05:32

Problem 5

Approximate the area under the curve $y=x^{3}-x$ from 0 to 1 using five equal subintervals.

Willis James
Willis James
Numerade Educator
05:03

Problem 6

Approximate the area under the curve $y=x^{2}-x$ from 0 to 1 using six equal subintervals.

In Problems 7 and 8, you will use Riemann sums to prove that $\int_{0}^{a} x^{2} d x=\frac{1}{3} a^{3} .$

Alayna Abraham
Alayna Abraham
Numerade Educator
03:16

Problem 7

. (a) Let the points $x_{0}=0<x_{1}<x_{2}<\cdots<x_{n}=a$ divide the interval into $n$ rectangles of equal width $w$. What is the width of each rectangle $w ?$
(b) Show that the total area of the rectangles used to approximate the area under $y=x^{2}$ between $x=0$ and $x=a$ is:
$$
S_{n}=\frac{a^{3}}{n^{3}}\left(1^{2}+2^{2}+3^{2}+\cdots+(n-1)^{2}\right)
$$
(c) Assume (the proof is worked out in Problem 8) that $1^{2}+2^{2}+$ $3^{2} \cdots+k^{2}=\frac{k(k+1)(2 k+1)}{6} .$ Use this formula to show that your expression for $S_{n}$ can be rewritten as:
$$
S_{n}=\frac{a^{3}}{6}\left(1-\frac{1}{n}\right)\left(2-\frac{1}{n}\right)
$$
(d) Evaluate the limit: $\lim _{n \rightarrow \infty} S_{n}$, and derive the formula for $\int_{0}^{a} x^{2} d x$

Fuzail Shakir
Fuzail Shakir
Numerade Educator
00:48

Problem 8

To prove the sum formula that we used in Problem 7 , namely:
$$
1^{2}+2^{2}+3^{2}+\cdots+k^{2}=\frac{k(k+1)(2 k+1)}{6}
$$
we will consider two sums:
$$
T_{k}=2^{3}+3^{3}+4^{3}+\cdots+k^{3}+(k+1)^{3}
$$
and
$$
U_{k}=1^{3}+2^{3}+3^{3}+\cdots+k^{3}
$$
Notice that these sums have exactly the same number of terms.
(a) Explain, by comparing which terms show up in both sums, why:
$$
T_{k}-U_{k}=(k+1)^{3}-1^{3}
$$
(b) Form the difference between the two sums by subtracting the first term of $U_{k}$ from the first term of $T_{k}$, the second term of $U_{k}$ from the second term of $T_{k}$, and so on. Then:

Linh Vu
Linh Vu
Numerade Educator
03:13

Problem 9

Approximate
$$
\int_{-1}^{1}\left(1-x^{2}\right) d x
$$
using five equal subintervals.

Goutam Chand
Goutam Chand
Numerade Educator
01:50

Problem 10

Approximate
$$
\int_{-1}^{1}\left(1+x^{2}\right) d x
$$
using five equal subintervals.

Vikash Ranjan
Vikash Ranjan
Numerade Educator
03:13

Problem 11

Approximate
$$
\int_{-1}^{1}\left(2+x^{2}\right) d x
$$
using five equal subintervals.

Goutam Chand
Goutam Chand
Numerade Educator
01:32

Problem 12

Approximate
$$
\int_{-2}^{2}\left(2+x^{2}\right) d x
$$
using six equal subintervals.

Tristan Williams
Tristan Williams
Numerade Educator
00:53

Problem 13

Approximate
$$
\int_{-1}^{2} e^{-x} d x
$$
using three equal subintervals.

Zachary Mitchell
Zachary Mitchell
Numerade Educator
01:05

Problem 14

Approximate
$$
\int_{0}^{\pi} \sin x d x
$$
using four equal subintervals.

Carson Merrill
Carson Merrill
Numerade Educator
03:33

Problem 15

. (a) Assume that $a, b>0$. Evaluate $\int_{a}^{b} x d x$, using the fact that the region bounded by $y=x$ and the $x$ -axis between $a$ and $b$ is a trapezoid. (See Figure 6.23.)

Gregory Higby
Gregory Higby
Numerade Educator
01:10

Problem 16

Assume that $a<0<b$. Use a geometric argument to show that
$$
\int_{a}^{b} x d x=\frac{b^{2}-a^{2}}{2}
$$
Express the limits in Problems 17-23 as definite integrals. In each case the n points $x_{0}<x_{1}<x_{2}<\cdots<x_{n}$ evenly divide the interval $[a, b]$ into n subintervals, each of width $w=(b-a) / n .$ You do not need to evaluate the definite integrals.

Lauren Shelton
Lauren Shelton
Numerade Educator
02:23

Problem 17

$$
\begin{array}{l}
\qquad \lim _{n \rightarrow \infty} w\left(\frac{x_{0}^{2}}{2}+\frac{x_{1}^{2}}{2}+\frac{x_{2}^{2}}{2}+\cdots+\frac{x_{n-1}^{2}}{2}\right) \\
\text { where } a=1, b=2
\end{array}
$$

Aayush Gupta
Aayush Gupta
Numerade Educator
02:03

Problem 18

$$
\begin{array}{l}
\lim _{n \rightarrow \infty} w\left(\sqrt{x_{0}+1}+\sqrt{x_{1}+1}+\sqrt{x_{2}+1}+\cdots+\sqrt{x_{n-1}+1}\right) \\
\text { where } a=0, b=3
\end{array}
$$

Hemraj Kumawat
Hemraj Kumawat
Numerade Educator
02:58

Problem 19

$$
\begin{array}{l}
\lim _{n \rightarrow \infty} w\left(\left(2 x_{0}-\frac{1}{2}\right)+\left(2 x_{1}-\frac{1}{2}\right)+\cdots+\left(2 x_{n-1}-\frac{1}{2}\right)\right) \\
\text { where } a=-1, b=1
\end{array}
$$

Abhijith V
Abhijith V
Numerade Educator
02:41

Problem 20

$$
\begin{array}{l}
\quad \lim _{n \rightarrow \infty} w\left(1+\frac{1}{x_{0}}\right)+\left(1+\frac{1}{x_{1}}\right)+\cdots+\left(1+\frac{1}{x_{n-1}}\right) \\
\text { where } a=1, b=5
\end{array}
$$

Malika Singh
Malika Singh
Numerade Educator
04:46

Problem 21

$$
\begin{array}{l}
\lim _{n \rightarrow \infty} w\left(2^{x_{0}}+2^{x_{1}}+2^{x_{2}}+\cdots+2^{x_{n-1}}\right) \\
\text { where } a=0 \text { and } b=1
\end{array}
$$

Malika Singh
Malika Singh
Numerade Educator
01:16

Problem 22

$$
\lim _{n \rightarrow \infty} w\left(\cos \left(x_{0}\right)+\cos \left(x_{1}\right)+\cdots+\cos \left(x_{n-1}\right)\right)
$$

AP
Aditya Panjiyar
Numerade Educator
02:43

Problem 23

$$
\begin{array}{l}
\lim _{n \rightarrow \infty} w\left(\frac{1}{e^{-x_{0}}+1}+\frac{1}{e^{-x_{1}}+1}+\cdots+\frac{1}{e^{-x_{n-1}}+1}\right) \\
\text { where } a=0 \text { and } b=2 \text { . }
\end{array}
$$

Abhijith V
Abhijith V
Numerade Educator
09:09

Problem 24

In Problems 24-29, express the definite integrals as limits of Riemann sums.
$$
\int_{-2}^{-1} \frac{x^{2}}{1+x^{2}} d x
$$

Ishita J.
Ishita J.
Numerade Educator
09:56

Problem 25

In Problems , express the definite integrals as limits of Riemann sum
$$
\int_{1}^{3}(x+1)^{1 / 3} d x
$$

Ishita J.
Ishita J.
Numerade Educator
07:43

Problem 26

In Problems , express the definite integrals as limits of Riemann sum
$$
\int_{1}^{3} e^{-2 x} d x
$$

Ishita J.
Ishita J.
Numerade Educator
04:59

Problem 27

In Problems , express the definite integrals as limits of Riemann sum
$$
\int_{1}^{e} \ln x d x
$$

Ishita J.
Ishita J.
Numerade Educator
01:46

Problem 28

In Problems , express the definite integrals as limits of Riemann sum
$$
\int_{0}^{\pi} \cos \frac{2 x}{\pi} d x
$$

Sahil Kumar
Sahil Kumar
Numerade Educator
View

Problem 29

In Problems , express the definite integrals as limits of Riemann sum
$$
\int_{0}^{5} x^{3} d x
$$

Suzanne W.
Suzanne W.
Numerade Educator
00:56

Problem 30

In Problems 30-36, use a graph to interpret the definite integral in terms of areas. Do not compute the integrals.
$$
\int_{0}^{5} e^{-x} d x
$$

Melissa Munoz
Melissa Munoz
Numerade Educator
01:16

Problem 31

In Problems , use a graph to interpret the definite integral in terms of areas. Do not compute the integrals.s.
$$
\int_{-1}^{2}\left(x^{2}-1\right) d x
$$

Adrian Co
Adrian Co
Numerade Educator
00:56

Problem 32

In Problems , use a graph to interpret the definite integral in terms of areas. Do not compute the integrals.s.$$
\int_{-2}^{2} \frac{1}{2} x^{3} d x
$$

Melissa Munoz
Melissa Munoz
Numerade Educator
00:56

Problem 33

In Problems , use a graph to interpret the definite integral in terms of areas. Do not compute the integrals.s.
$$
\int_{0}^{3}(2 x+1) d x
$$

Melissa Munoz
Melissa Munoz
Numerade Educator
01:07

Problem 34

In Problems , use a graph to interpret the definite integral in terms of areas. Do not compute the integrals.s.
$$
\int_{-\pi}^{\pi} \cos x d x
$$

Amy Jiang
Amy Jiang
Numerade Educator
00:56

Problem 35

In Problems , use a graph to interpret the definite integral in terms of areas. Do not compute the integrals.s.
$$
\int_{-3}^{2}\left(1-\frac{1}{2} x\right) d x
$$

Melissa Munoz
Melissa Munoz
Numerade Educator
00:56

Problem 36

In Problems , use a graph to interpret the definite integral in terms of areas. Do not compute the integrals.s.
$$
\int_{1 / 2}^{4} \ln x d x
$$

Melissa Munoz
Melissa Munoz
Numerade Educator
01:02

Problem 37

In Problems $37-47$, use an area formula from geometry to find the value of each integral by interpreting it as the (signed) area under the graph of an appropriately chosen function.
$$
\int_{-2}^{3}|x| d x
$$

Linh Vu
Linh Vu
Numerade Educator
01:11

Problem 38

In Problems , use an area formula from geometry to find the value of each integral by interpreting it as the (signed) area under the graph of an appropriately chosen function.
$$
\int_{-3}^{3} \sqrt{9-x^{2}} d x
$$

Adrian Co
Adrian Co
Numerade Educator
01:31

Problem 39

In Problems , use an area formula from geometry to find the value of each integral by interpreting it as the (signed) area under the graph of an appropriately chosen function.
$$
\int_{2}^{5}\left(\frac{1}{2} x-4\right) d x
$$

Adrian Co
Adrian Co
Numerade Educator
01:23

Problem 40

In Problems , use an area formula from geometry to find the value of each integral by interpreting it as the (signed) area under the graph of an appropriately chosen function.
$$
\int_{1 / 2}^{1} \sqrt{1-x^{2}} d x
$$

Adrian Co
Adrian Co
Numerade Educator
01:15

Problem 41

$$
\int_{-2}^{2}\left(\sqrt{4-x^{2}}-2\right) d x
$$

John Nicolle
John Nicolle
Numerade Educator
01:23

Problem 42

In Problems , use an area formula from geometry to find the value of each integral by interpreting it as the (signed) area under the graph of an appropriately chosen function.
$$
\int_{0}^{1} \sqrt{2-x^{2}} d x
$$

Adrian Co
Adrian Co
Numerade Educator
04:04

Problem 43

In Problems , use an area formula from geometry to find the value of each integral by interpreting it as the (signed) area under the graph of an appropriately chosen function.
$$
\int_{2}^{0}\left(4-\sqrt{9-x^{2}}\right) d x
$$

Priyanka Sadarangani
Priyanka Sadarangani
Numerade Educator
02:18

Problem 44

In Problems , use an area formula from geometry to find the value of each integral by interpreting it as the (signed) area under the graph of an appropriately chosen function.
$$
\int_{-2}^{1} \sqrt{4-x^{2}} d x
$$

JC
Jeff Christopher
Numerade Educator
06:31

Problem 45

In Problems , use an area formula from geometry to find the value of each integral by interpreting it as the (signed) area under the graph of an appropriately chosen function.
$$
\int_{-1}^{2}(2-|x|) d x
$$

Ahmad Reda
Ahmad Reda
Numerade Educator
01:26

Problem 46

In Problems , use an area formula from geometry to find the value of each integral by interpreting it as the (signed) area under the graph of an appropriately chosen function.
$$
\int_{0}^{2}|x-1| d x
$$

Adrian Co
Adrian Co
Numerade Educator
00:19

Problem 47

In Problems , use an area formula from geometry to find the value of each integral by interpreting it as the (signed) area under the graph of an appropriately chosen function.
$$
\int_{0}^{2}(2+x) d x
$$

Victoria Dollar
Victoria Dollar
Numerade Educator
03:46

Problem 48

Use the definition of the Riemann integral in terms of Riemann sums to prove property (3) of definite integrals. That is, if $f(x)$ is continuous on $[a, b]$ and $k$ is any constant, then:
$$
\int^{b} k f(x) d x=k \int^{b} f(x) d x
$$

Shafiq Rehman
Shafiq Rehman
Numerade Educator
04:06

Problem 49

Use a diagram to explain why, if $f(x)$ is continuous on an interval that contains all of the points $a, b, c$, then
$$
\int_{a}^{b} f(x) d x=\int_{a}^{c} f(x) d x+\int_{c}^{b} f(x) d x
$$
That is, derive property (5) of definite integrals. You should consider the cases
(a) $b<a<c$
(b) $c<b<a$.

Priyanka Sadarangani
Priyanka Sadarangani
Numerade Educator
01:42

Problem 50

Given that $\int_{0}^{u} x^{2} d x=\frac{1}{3} a^{3}$ evaluate the following:
(a) $\int_{0}^{1} \frac{1}{2} x^{2} d x$
(b) $\int_{0}^{-1} 3 x^{2} d x$
(c) $\int_{-1}^{2} \frac{1}{3} x^{2} d x$
(d) $\int_{1}^{1} 3 x^{2} d x$
(e) $\int_{-2}^{3}(x+1)^{2} d x$
(f) $\int_{2}^{4}(x-2)^{2} d x$

Sriparna Bhattacharjee
Sriparna Bhattacharjee
Numerade Educator
01:02

Problem 51

$$
\text { Find } \int_{2}^{2} \cos \left(3 x^{2}\right) d x
$$

Nick Johnson
Nick Johnson
Numerade Educator
01:49

Problem 52

$$
\text { Find } \int_{-3}^{-3} e^{-x^{2} / 2} d x \text { . }
$$

Nick Johnson
Nick Johnson
Numerade Educator
01:10

Problem 53

$$
\text { Find } \int_{-1}^{1} 3 x d x \text { . }
$$

Amy Jiang
Amy Jiang
Numerade Educator
01:23

Problem 54

$$
\text { Find } \int_{-1}^{1} 3 x^{5} d x
$$

Nishant Tyagi
Nishant Tyagi
Numerade Educator
01:57

Problem 55

$$
\text { Find } \int_{0}^{2}(x-1)^{3} d x \text { . }
$$

Shivani Yadav
Shivani Yadav
Numerade Educator
01:46

Problem 56

Explain geometrically why
$$
\int_{1}^{2} x^{2} d x=\int_{0}^{2} x^{2} d x-\int_{0}^{1} x^{2} d x
$$
and show that (6.3) can be written as
$$
\int_{1}^{2} x^{2} d x=\int_{1}^{0} x^{2} d x+\int_{0}^{2} x^{2} d x
$$

Abigail Darko
Abigail Darko
Numerade Educator
01:37

Problem 57

Given that $\int_{0}^{a} x^{3} d x=\frac{1}{4} a^{4}$, evaluate the following integrals:
(a) $\int_{0}^{2} x^{3} d x$
(b) $\int_{0}^{1} 2 x^{3} d x$
(c) $\int_{-1}^{1} 2 x^{3} d x$
(d) $\int_{-1}^{1}(x+1)^{3} d x$
(e) $\int_{1}^{2} 2(x+2)^{3} d x$.

Adrian Co
Adrian Co
Numerade Educator
02:19

Problem 58

Given that $\int_{0}^{a} x^{4} d x=\frac{1}{5} a^{5}$ evaluate the following integrals
(a) $\int_{0}^{2} x^{4} d x$
(b) $\int_{0}^{1} \frac{x^{4}}{2} d x$
(c) $\int_{-1}^{1} \frac{x^{4}}{2} d x$
(d) $\int_{-2}^{0}(x+2)^{4} d x$
(e) $\int_{-3}^{0}(x+1)^{4} d x$
(f) $\int_{0}^{2} 2(x-2)^{4} d x$.

Linh Vu
Linh Vu
Numerade Educator
00:57

Problem 59

In Problems 59-63, verify each inequality without evaluating the integrals.
$$
\int_{0}^{1} x d x \geq \int_{0}^{1} x^{2} d x
$$

Nick Johnson
Nick Johnson
Numerade Educator
00:46

Problem 60

In Problems , verify each inequality without evaluating the integrals.
$$
\int_{2}^{4} x d x \leq \int_{2}^{4} x^{2} d x
$$

Nick Johnson
Nick Johnson
Numerade Educator
00:32

Problem 61

In Problems , verify each inequality without evaluating the integrals.
$$
0 \leq \int_{0}^{9} \sqrt{x} d x \leq 27
$$

Amrita Bhasin
Amrita Bhasin
Numerade Educator
02:12

Problem 62

In Problems , verify each inequality without evaluating the integrals.
$$
\sqrt{3} \leq \int_{0}^{1} \sqrt{4-x^{2}} d x \leq 2
$$

Linda Hand
Linda Hand
Numerade Educator
View

Problem 63

In Problems , verify each inequality without evaluating the integrals.
$$
\frac{\pi}{3} \leq \int_{\pi / 6}^{5 \pi / 6} \sin x d x \leq \frac{2 \pi}{3}
$$

Suzanne W.
Suzanne W.
Numerade Educator
02:59

Problem 64

$$
\text { Find the value of } a \geq 0 \text { that maximizes } \int_{0}^{a}\left(4-x^{2}\right) d x \text { . }
$$

Sam Sohn
Sam Sohn
Numerade Educator
06:36

Problem 65

$$
\text { Find the value of } a \in[0,2 \pi] \text { that maximizes } \int_{0}^{a} \cos x d x \text { . }
$$

Uma Kumari
Uma Kumari
Numerade Educator
03:00

Problem 66

$$
\text { Find } a \in(0,2 \pi] \text { such that } \int_{0}^{a} \sin x d x=0
$$

M Hassan Anwar
M Hassan Anwar
Numerade Educator
02:28

Problem 67

$$
\text { Find } a>1 \text { such that } \int_{1}^{a}(x-3)^{3} d x=0
$$

Melinda Mulcahy
Melinda Mulcahy
Numerade Educator
01:03

Problem 68

$$
\text { Find } a>0 \text { such that } \int_{0}^{a}(1-x) d x=0
$$

Lauren Shelton
Lauren Shelton
Numerade Educator
03:06

Problem 69

Total Rainfall A rain gauge is set up to measure the amount of rainfall occurring in $1 \mathrm{hr}$ on the UCLA campus (the readout
from the rain gauge is in $\mathrm{mm} / \mathrm{hr}$ ). Assume that the following data is collected in a 6 hour window.
$$
\begin{array}{c|c}
\hline \text { Time, } t & \text { Rainfall rate, } \boldsymbol{r}(\boldsymbol{t}) \text { in } \mathrm{mm} / \mathrm{hr} \\
\hline 0 & 1 \\
1 & 2 \\
2 & 3 \\
3 & 1 \\
4 & 1 \\
5 & 0 \\
6 & 0 \\
\hline
\end{array}
$$

Eric Mockensturm
Eric Mockensturm
Numerade Educator
03:30

Problem 70

You are measuring the ability of an antibiotic to kill harmful bacteria. You measure the rate at which the antibiotic kills bacteria (i.e., number of bacteria killed in one hour); this is called the mortality rate. You measure the following data for the number of bacteria killed in a 12 hour time period starting at $t=0$, and ending at $t=12$.
$$
\begin{array}{c|c}
\hline \text { Time, } t & \text { Mortality rate, per hour } \boldsymbol{m}(\boldsymbol{t}) \\
\hline 0 & 20 \\
1 & 300 \\
2 & 350 \\
3 & 400 \\
4 & 500 \\
5 & 450 \\
6 & 410 \\
7 & 350 \\
8 & 320 \\
9 & 300 \\
10 & 200 \\
11 & 100 \\
12 & 110 \\
\hline
\end{array}
$$
(a) Use six even subintervals to approximate the total number of deaths between $t=0$ and $t=6$ and evaluate this sum using the data in the table.
(b) Use six even subintervals to approximate the total number of deaths between $t=0$ and $t=12$ and evaluate this sum using the data in the table.
(c) Use four even subintervals to approximate the total number of deaths between $t=4$ and $t=12$ and evaluate this sum using the data in the table.

Carson Merrill
Carson Merrill
Numerade Educator