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C HAPTER
6 SE Applications of
Integration - TI
Chapter Outline
6.1 THEAREABETWEEN CURVES
www.ck12.org Chapter 6. SE Applications of Integration - TI
6.1 The Area Between Curves
This activity is intended to supplement Calculus, Chapter 5, Lesson 1.
In this activity, you will explore:
• Using integrals to find the area between two curves.
Use this document to record your answers.
Problem 1 –Making Sidewalks
While integrals can be used to find the area under a curve, they can also be used to find the area between curves through subtraction (just make sure the subtraction order is the top curve minus bottom curve.)
Suppose you are a building contractor and need to know how much concrete to order to create a pathway that is 13 foot deep. To find the volume, all that is needed is to multiply the area of the sidewalk by the depth of the sidewalk.
The sidewalk borders can be modeled from−2πto 2πby:
f(x) =sin(0.5x) +3 andg(x) =sin(0.5x).
Graph both functions. Adjust the window settings to−6.4≤x≤6.4 and−5≤y≤5.
Now use the Integral tool from the Math menu to calculate the integrals of f(x)andg(x).
Enter−2πfor the lower limit and 2πfor the upper limit.
• What is the value of the integral of f(x)? Ofg(x)?
On the Home screen, define f(x)andg(x). Then use the Numerical Integral command (nInt) in the Calc menu to find the area of the pathway.
Hint:the area is equal to the integral of f(x)−g(x).
Note: ThenIntcommand has the syntax:nInt(function, variable, left limit, right limit)
• What is the formula for the volume of the sidewalk?
• Now calculate how much concrete is needed for the pathway.
Problem 2 –Finding New Pathways
The owners have changed the design of the pathway. It will now be modeled from -2 to 2 by:
f(x) =x(x+2.5)(x−1.5) +3 g(x) =x(x+2)(x−2)
Graph both functions. Adjust the window settings to−4≤x≤4 and−5≤y≤9.
Calculate the integrals of f(x)andg(x). Enter−2 for the lower limit and 2 for the upper limit.
6.1. The Area Between Curves www.ck12.org
• What is the value of the integral of f(x)? Ofg(x)?
• Now calculate how much concrete is needed for the pathway on the Home screen. Remember to define f(x) andg(x).
Problem 3 –Stepping Stones
The owners also want stepping stones, which can be modeled by
f(x) =−(x−1)(x−2) +2 g(x) = (x−1)(x−2) +0.5.
This situation different because the starting and stopping points are not given. Assume that the stepping stones are
1
3 foot thick.
Graph both functions. Adjust the window settings to−1≤x≤7 and−4≤y≤4 with a step size of 0.5 for both.
Use theIntersectiontool in the Math menu to find the intersection points. You can also use theSolvecommand on the Home screen.
• What are the coordinates for the two intersection points?
Calculate the integrals of f(x)andg(x). Use thex−values of the intersection points as the lower and upper limits.
• What is the value of the integral of f(x)? g(x)?
• Now calculate how much concrete is needed for the pathway on the Home screen. Remember to define f(x) andg(x).
www.ck12.org Chapter 6. SE Applications of Integration - TI
6.2 Volume by Cross-Sections
This activity is intended to supplement Calculus, Chapter 5, Lesson 2.
Part 1 –Setting Up The Problem And Understanding The Concept
A first step to solve calculus volume problems is to label the point and differential. You know the volume of an object is the area of the base times its height. So the differentialdV equals area·dxor area·dy.
1. Typically the cross section is perpendicular to an axis. If the shape formed is perpendicular to thex−axis, what is the differential?
2. The function may define the base with cross sections that form a variety of shapes.
a. What is the area of a square?
b. What is the area of a semicircle?
3. Consider a function that defines the base of a solid where the cross sections perpendicular to the x−axis form equilateral triangles. Let the base of the triangle be parallel to they−axis. What is the area of the triangle? Draw a sketch and justify your answer.
4. If the length of one of the sides of this equilateral triangle is 1 cm, calculate the area. Show your calculation.
5. Let the first quadrant region enclosed by the graph ofy= √
x·e−x2 and the linex=2 be the base of a solid. If the cross sections perpendicular to thex−axis are equilateral triangles, what is the volume of the solid? Show your work.
6.2. Volume by Cross-Sections www.ck12.org
Part 2 –Homework
Questions 1 and 2 are non-calculator, exam-like problems. Show all your work. On Question 3, just show the set up and then use your calculator to find your answer.
1. Let the first quadrant region enclosed by the graph ofy=1x and the linex=1 andx=4 be the base of a solid. If the cross sections perpendicular to thexaxis are semicircles, what is the volume of the solid? Show your work.
2. Let the base of a solid be the first quadrant region enclosed by thex−axis, they−axis and the graph ofy=1−x42. If all the cross sections perpendicular to they−axis are squares, what is the volume of the solid? Show your work.
3. Let the base of a solid be the first quadrant region enclosed by thex−axis and one arch of the graphy=sin(x). If all cross sections perpendicular to thex−axis are squares, then approximately what is the volume of the solid? Show your set up.
www.ck12.org Chapter 6. SE Applications of Integration - TI
6.3 Gateway Arc Length
This activity is intended to supplement Calculus, Chapter 5, Lesson 3.
Part 1 –Arc Length Introduced
The Gateway to the West is an inverted catenary arch in St. Louis that is about 630 feet tall and 630 feet wide at its base. A catenary (a hyperbolic cosine function) is the shape that a chain or cable forms when it hangs between two points. The shape of the Gateway Arch can be modeled by the following equation:
y(x) =−68.8 cosh(0.01x−3) +700.
1. If you were to ride in the elevator tram of the Gateway Arch, you would travel at least how far to get to the top?
Explain.
Using the Pythagorean Theorem, we getdL= s
dx dx
2
+ dy
dx 2
dx. Asdxbecomes smaller, so dodyanddL.
As dL becomes smaller, the difference in length of dL and the length of the arc from x to x+dx is eventually infinitesimal. So we can integrate both sides to give us the formula for arc length:
L=
b
Z
a
s
1+ dy
dx 2
dx.
2. On the Home screen, use the formula to find the arc length fromx=0 tox=300 fory(x) =−68.8 cosh(0.01x− 3) +700. Write the formula and answer. Is this reasonable (when compared to your answer from Exercise 1)?
For parametric equations, the Pythagorean Theorem would yield dL= s
dx dt
2
+ dy
dt 2
dt. Integrating both
sides gives us the arc length formulaL=
b
R
a
q
(x0(t))2+ (y0(t))2
dt.
6.3. Gateway Arc Length www.ck12.org Graph the parametric equationsx(t) =2 cos(t)andy(t) =2 sin(t).
3. For the parametric equationx(t) =2 cos(t)andy(t) =2 sin(t), use the arc length formula to find the length from t=0 tot=π2 Show each step.
Now graph the equationy1(x) = p
4−x2. Whenx=0 tox=2, this graph should look the same as the previous parametric curve.
4. Use the Home screen to find the arc length ofy1(x) = p
4−x2fromx=0 tox=2. Write out the equation and answer. Does this agree with the previous answer? Why or why not?
5. Graphy2(x) =x2−9 and approximate the arc length fromx=0 to x=3. Write the arc length formula and solution for this arc length. Try usingarcLen(y2(x),x,0,3)on the Home screen to check your answer.
6. Use the Pythagorean Theorem to approximate the arc length fromx=0 tox=3 ofy=−x2+53x+4. On the Home screen, find the arc length using the formula. Write the formula and solution. Discuss if this is reasonable.
Part 2 –Additional Practice
1. Which of the following integrals gives the length of the graph of y=arcsinx2 betweenx=aandx=b, where 0<a<b<π2 ?
a.
b
R
a
s x2+8 x2+4dx
b.
b
R
a
s x2+6 x2+4dx c.
b
R
a
s x2−2 x2−4dx
d.
b
R
a
s x2−5 x2−4dx e.
b
R
a
s 2x2+3
x2+1 dx
2. The length of the curve determined by the parametric equationsx=sintandy=tfromt=0 tot=πis a.
π
R
0
pcos2t+1dt b.
Rπ
0
p
sin2t+1dt c.
π
R
0
√
cost+1dt d.
π
R
0
√
sint+1dt e.
Rπ
0
√
1−cost dt
3. Which of the following integrals gives the length of the graph ofy=tan x betweenx=a and x=b, where 0<a<b<π2 ?
www.ck12.org Chapter 6. SE Applications of Integration - TI a.
b
R
a
px2+tan2x dx
b.
b
R
a
√x+tanx dx
c.
b
R
a
p1+sec2x dx d.
b
R
a
p1+tan2x dx
e.
b
R
a
p1+sec4x dx
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C HAPTER
7 SE Transcendental
Functions - TI
Chapter Outline
7.1 INVERSES OFFUNCTIONS