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.
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Riemann Sum
Riemann Sums is a special type of approximation of the area under a curve by dividing it into multiple simple shapes like rectangles or trapezoids and is used in integrals when finite sums are involved. Figuring out the area of a curve is complex hence this method makes it simple. Usually, we take the help of different integration methods for this purpose. This is one of the major parts of integral calculus.
Riemann Integral
Bernhard Riemann's integral was the first systematic description of the integral of a function on an interval in the branch of mathematics known as real analysis.
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![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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