Chapter 6, Applications of Integrals Video Solutions, Biocalculus Calculus for the Life Sciences

02:46

Problem 1

Find the area of the shaded region.

Katelyn Chen

Numerade Educator

00:47

Problem 2

Find the area of the shaded region.

Fuzail Shakir

Numerade Educator

04:06

Sketch the region enclosed by the given curves. Draw a typical approximating rectangle and label its height and width. Then find the area of the region.
$y=e^{x}, \quad y=x^{2}-1, \quad x=-1, \quad x=1$

Sky Li

Numerade Educator

00:09

Problem 4

Sketch the region enclosed by the given curves. Draw a typical approximating rectangle and label its height and width. Then find the area of the region.
$y=\ln x, \quad x y=4, \quad x=1, \quad x=3$

Fuzail Shakir

Numerade Educator

09:51

Problem 5

Sketch the region enclosed by the given curves. Draw a typical approximating rectangle and label its height and width. Then find the area of the region.
$y=x^{2}, \quad y^{2}=x$

Sky Li

Numerade Educator

01:07

Problem 6

Sketch the region enclosed by the given curves. Draw a typical approximating rectangle and label its height and width. Then find the area of the region.
$y=x^{2}-2 x, \quad y=x+4$

Fuzail Shakir

Numerade Educator

10:16

Problem 7

Sketch the region enclosed by the given curves and find its area.
$y=12-x^{2}, \quad y=x^{2}-6$

Ian Musser

Numerade Educator

01:01

Problem 8

Sketch the region enclosed by the given curves and find its area.
$y=x^{2}, \quad y=4 x-x^{2}$

Fuzail Shakir

Numerade Educator

10:21

Problem 9

Sketch the region enclosed by the given curves and find its area.
$y=e^{x}, \quad y=x e^{x}, \quad x=0$

Sinisa Stura

Numerade Educator

00:43

Problem 10

Sketch the region enclosed by the given curves and find its area.
$y=\cos x, \quad y=2-\cos x, \quad 0 \leqslant x \leqslant 2 \pi$

Fuzail Shakir

Numerade Educator

13:50

Problem 11

Use a graph to find approximate $x$ -coordinates of the points of intersection of the given curves. Then find (approximately) the area of the region bounded by the curves.
$y=x \sin \left(x^{2}\right), \quad y=x^{4}$

Sicelo Goqo

Numerade Educator

01:22

Problem 12

Use a graph to find approximate $x$ -coordinates of the points of intersection of the given curves. Then find (approximately) the area of the region bounded by the curves.
$y=x \cos x, \quad y=x^{10}$

Fuzail Shakir

Numerade Educator

03:13

Problem 13

Sketch the region that lies between the curves $y=\cos x$ and $y=\sin 2 x$ and between $x=0$ and $x=\pi / 2 .$ Notice that the region consists of two separate parts. Find the area of this region.

Bailey Brooks

Numerade Educator

00:38

Problem 14

Sketch the curves $y=\cos x$ and $y=1-\cos x$ $0 \leqslant x \leqslant \pi,$ and observe that the region between them consists of two separate parts. Find the area of this region.

Fuzail Shakir

Numerade Educator

08:38

Problem 15

Sometimes it's easier to find an area by regarding $x$ as a function of $y$ instead of $y$ as a function of $x .$ To illustrate this idea, let $S$ be the region enclosed by the line $y=x-1$ and the parabola $y^{2}=2 x+6$
(a) By sketching $S$ , observe that if you want to integrate with respect to $x$ you have to split $S$ into two parts with different boundary curves.
(b) If you integrate with respect to $y,$ observe that there is a left boundary curve and a right boundary curve.
(c) Find the area of S using the method of either part (a) or part (b).

David Marsella

Numerade Educator

01:07

Problem 16

Find the area of the region enclosed by the curves $y=x$ and $4 x+y^{2}=12$

Fuzail Shakir

Numerade Educator

03:06

Problem 17

A laurel leaf is shown. Estimate its area using the Midpoint Rule with six subintervals.

Gregory Higby

Numerade Educator

01:10

Problem 18

A cicada wing is shown. Estimate its area using the Midpoint Rule with six subintervals.

Fuzail Shakir

Numerade Educator

06:37

Problem 19

A cross-section of an airplane wing is shown. Measurements of the thickness of the wing, in centimeters, at 20 -centimeter intervals are $5.8,20.3,26.7,29.0,27.6,27.3,$ $23.8,20.5,15.1,8.7,$ and $2.8 .$ Use the Midpoint Rule to estimate the area of the wing's cross-section.

Lucas Everson

Numerade Educator

01:05

Problem 20

The widths (in meters) of a kidney-shaped swimming pool were measured at 2 - meter intervals as indicated in the figure. Use the Midpoint Rule to estimate the area of the
pool.

Fuzail Shakir

Numerade Educator

11:56

Problem 21

Cerebral blood flow The table shows measurements of $A(t),$ the concentration of $\mathrm{N}_{2} \mathrm{O}$ flowing into a patient's brain, and $V(t),$ the concentration of $\mathrm{N}_{2} \mathrm{O}$ flowing out of the brain, where $t$ is measured in minutes and $A(t)$ and $V(t)$ are measured in mL of $\mathrm{N}_{2} \mathrm{O}$ per mL of blood.
$$\begin{array}{|c|c|c|}\hline t & {A(t)} & {V(t)} \\ \hline 1 & {0.031} & {0.008} \\ {3} & {0.041} & {0.029} \\ {5} & {0.042} & {0.035} \\ {7} & {0.044} & {0.042} \\ {9} & {0.045} & {0.044} \\ \hline\end{array}$$
(a) Use the Midpoint Rule to estimate $\int_{0}^{10}[A(t)-V(t)] d t$
(b) If the volume of $\mathrm{N}_{2} \mathrm{O}$ absorbed by the brain in the first 10 minutes is $64 \mathrm{mL},$ calculate the cerebral blood flow.

Ali Crampton

Numerade Educator

00:39

Problem 22

Cerebral blood flow Models for the arterial and venous
concentration functions in Figure 8 are given by
$$\quad A(t)=\frac{0.05 t^{2}}{t^{2}+1} \quad V(t)=\frac{0.05 t^{2}}{t^{2}+7}$$
(a) Find the area between the graphs of $A$ and $V$ for 0$\leqslant t \leqslant 10$ .
(b) If the volume of $\mathrm{N}_{2} \mathrm{O}$ absorbed by the brain in the first 10 minutes is $60 \mathrm{mL},$ determine the cerebral blood flow.

Fuzail Shakir

Numerade Educator

01:28

Problem 23

Racing cars driven by Chris and Kelly are side by side at the start of a race. The table shows the velocities of each car (in miles per hour) during the first 10 seconds of the race. Use the Midpoint Rule to estimate how much farther Kelly travels than Chris does during the first 10 seconds.
$$\begin{array}{|c|c|c|}\hline t & {v_{c}} & {v_{K}} \\ \hline 0 & {0} & {0} \\ {1} & {20} & {22} \\ {2} & {32} & {37} \\ {3} & {46} & {52} \\ {4} & {54} & {61} \\ {5} & {62} & {71} \\ \hline\end{array}
\begin{array}{|c|c|c|}\hline t & {v_{C}} & {v_{K}} \\ \hline 6 & {69} & {80} \\ {7} & {75} & {86} \\ {8} & {81} & {93} \\ {9} & {86} & {98} \\ {10} & {90} & {102} \\ \hline\end{array}
$$

James Kiss

Numerade Educator

00:41

Problem 24

Two cars, $A$ and $B,$ start side by side and accelerate from rest. The figure shows the graphs of their velocity functions.
(a) Which car is ahead after one minute? Explain.
(b) What is the meaning of the area of the shaded region?
(c) Which car is ahead after two minutes? Explain.
(d) Estimate the time at which the cars are again side by side.

Fuzail Shakir

Numerade Educator

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Problem 25

Birth and death rates If the birth rate of a population is $b(t)=2200 e^{0.04 t}$ people per year and the death rate is $d(t)=1460 e^{0.018 t}$ people per year, find the area between
these curves for 0$\leqslant t \leqslant 10 .$ What does this area represent?