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Problem: Use the definition of the Riemann integral to prove that:
The problem image is attached.
You will need to use the definitions in the definition image (attached) to solve the problem.
Transcribed Image Text:f(x) = 3x + 2 is Riemann integrable on [0, 3], and f f (x)dx = ³/2
39
2
Transcribed Image Text:Definition. Let f:[a,b] → R be a bounded function and P be a partition of [a,b]
Define
Ax₁ = x₁ - x₁-1
M₁(ƒ) = sup{ƒ(x) : x = [x;_, , x; ]}
m, (f) = inf{f(x): x € [x₁,_,,x,]}
U(P,ƒ) = [M,(ƒ)^x,
L(P,ƒ) = ±m, (ƒ)Ax,
i=1
U(ƒ) = inf{U(P, f):P is a partition of [a, b]}
L(f) = sup{L(P, f): P is a partition of [a, b]}
Definition. Let ƒ : [a,b] → R be a bounded function. Then f is a Riemann Integrable
on [a, b] if U(ƒ) = L(ƒ) . If ƒ is a Riemann Integrable on [a, b] them we write
b
ƒ ƒ(x)dx = U(ƒ) = L(ƒ)
a
With differentiation, one of the major concepts of calculus. Integration involves the calculation of an integral, which is useful to find many quantities such as areas, volumes, and displacement.
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![f(x) = 3x + 2 is Riemann integrable on [0, 3], and f f (x)dx = ³/2
39
2](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe00cebfb-aec4-474d-8979-79a2d105b819%2F3b13a805-fba5-42f5-a3f5-11120d9b381c%2F0w0bofc_processed.jpeg&w=3840&q=75)
![Definition. Let f:[a,b] → R be a bounded function and P be a partition of [a,b]
Define
Ax₁ = x₁ - x₁-1
M₁(ƒ) = sup{ƒ(x) : x = [x;_, , x; ]}
m, (f) = inf{f(x): x € [x₁,_,,x,]}
U(P,ƒ) = [M,(ƒ)^x,
L(P,ƒ) = ±m, (ƒ)Ax,
i=1
U(ƒ) = inf{U(P, f):P is a partition of [a, b]}
L(f) = sup{L(P, f): P is a partition of [a, b]}
Definition. Let ƒ : [a,b] → R be a bounded function. Then f is a Riemann Integrable
on [a, b] if U(ƒ) = L(ƒ) . If ƒ is a Riemann Integrable on [a, b] them we write
b
ƒ ƒ(x)dx = U(ƒ) = L(ƒ)
a](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe00cebfb-aec4-474d-8979-79a2d105b819%2F3b13a805-fba5-42f5-a3f5-11120d9b381c%2Fcys6l84_processed.jpeg&w=3840&q=75)
