1. (a) Use Section 6.2 (Region between Curves) in the textbook to compute the area of 6 - y, and the region R in the first quadrants bounded by the x axis, the line x = the curve y = Vx. (b) Now use the double integral to compute the area of R above.
1. (a) Use Section 6.2 (Region between Curves) in the textbook to compute the area of 6 - y, and the region R in the first quadrants bounded by the x axis, the line x = the curve y = Vx. (b) Now use the double integral to compute the area of R above.
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter9: Systems Of Equations And Inequalities
Section: Chapter Questions
Problem 12T
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Riemann Integral
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Question
![6.2 Regions Between Curves
In this section, the method for finding the area of a region bounded by a single curve is
generalized to regions bounded by two or more curves. Consider two functions f and
g continuous on an interval [a, b] on which f(x) = g(x) (Figure 6.11). The goal is to
find the area A of the region bounded by the two curves and the vertical lines x = a and
x = b.
Once again, we rely on the slice-and-sum strategy (Section 5.2) for finding areas
by Riemann sums. The interval [a, b] is partitioned into n subintervals using uniformly
spaced grid points separated by a distance Ax = (b – a)/n (Figure 6.12). On each
subinterval, we build a rectangle extending from the lower curve to the upper curve. On
the kth subinterval, a point x is chosen, and the height of the corresponding rectangle is
taken to be f(x;) – 8(x;). Therefore, the area of the kth rectangle is (f(x;) – g(x;))Ax
(Figure 6.13). Summing the areas of then rectangles gives an approximation to the area of
the region between the curves:
A = E(f(x;) – 8(x;))Ax.
k-1
(xị. f(x))
Area of kth rectangle
= (S() – g(x)) Ar
y = f(x)
f(x) - 8(x)
y = f(x)
a
Ar
b
b = x.
y = g(x)
y = g(x)
Ax = width of each rectangle
Area of region: A =
Fep - g(x)) Ax
Figure 6.12
Figure 6.13
As the number of grid points increases, Ax approaches zero and these sums approach the
area of the region between the curves; that is,
A = lim (f(x;) – g(x;))Ax.
n* k-1
The limit of these Riemann sums is a definite integral of the function f – g.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Faba5746d-e0e8-46ca-a0bb-ec14000055d1%2F8b697f47-f7bf-466c-a0ef-e4961928562b%2Fngpd2oj_processed.png&w=3840&q=75)
Transcribed Image Text:6.2 Regions Between Curves
In this section, the method for finding the area of a region bounded by a single curve is
generalized to regions bounded by two or more curves. Consider two functions f and
g continuous on an interval [a, b] on which f(x) = g(x) (Figure 6.11). The goal is to
find the area A of the region bounded by the two curves and the vertical lines x = a and
x = b.
Once again, we rely on the slice-and-sum strategy (Section 5.2) for finding areas
by Riemann sums. The interval [a, b] is partitioned into n subintervals using uniformly
spaced grid points separated by a distance Ax = (b – a)/n (Figure 6.12). On each
subinterval, we build a rectangle extending from the lower curve to the upper curve. On
the kth subinterval, a point x is chosen, and the height of the corresponding rectangle is
taken to be f(x;) – 8(x;). Therefore, the area of the kth rectangle is (f(x;) – g(x;))Ax
(Figure 6.13). Summing the areas of then rectangles gives an approximation to the area of
the region between the curves:
A = E(f(x;) – 8(x;))Ax.
k-1
(xị. f(x))
Area of kth rectangle
= (S() – g(x)) Ar
y = f(x)
f(x) - 8(x)
y = f(x)
a
Ar
b
b = x.
y = g(x)
y = g(x)
Ax = width of each rectangle
Area of region: A =
Fep - g(x)) Ax
Figure 6.12
Figure 6.13
As the number of grid points increases, Ax approaches zero and these sums approach the
area of the region between the curves; that is,
A = lim (f(x;) – g(x;))Ax.
n* k-1
The limit of these Riemann sums is a definite integral of the function f – g.
Transcribed Image Text:1. (a) Use Section 6.2 (Region between Curves) in the textbook to compute the area of
6 – y, and
the region R in the first quadrants bounded by the x axis, the line x =
the curve y = Vx.
(b) Now use the double integral to compute the area of R above.
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