Assume that: f:[0,4/3] → R with f(x) = -0. 5 x² 1- Compute U(P,f), L(P, f) 2- compute 1/3 1/3 ! | f(x)dx, f(x)dx 3- show that the function f is R-integrable on| 0,1/3 and find the value of integral.

Q: Compute the integral of the scalar function f(x, y, z) = 5xy + 9z over r(t) = (cos (t), sin (t), t)…

A:

Q: Compute the double integral R ye®Y dA where Ris the square -1 < x < 1, 0 < y< 2.

A:

Q: B-Let f(x,y)=(y if x is rational, 1-y if x is irrational). Show that f(x,y)dydx = ? also prove that…

A:

Q: Y dA + x10 + y10 + 10 D over the domain D = [-3, 2] × [–-1, 1].

A:

Q: 1- Assume that x, y ER and z = x+ iy e C. Calculate the following integrals on C = {z: |z| = 1}…

A: Cauchy integral formula

Q: :. Evaluate the double integral (x + y)² sin(x – y) dA R for R, which is square with the vertices…

A:

Q: z dV, B (x – 1)2 onde B := {(r, y, x): (y – 1)² + 22 < 2z 4

A: Here is the solution of the given problem. Volume integral:

Q: (b) Show that 1/2 1/2 9. 1/2 [f(x)+ g(x)]² dx, g (x) dx a This is the Minkowski inequality for…

A:

Q: Define h(0,0) in such a way that h is continuous at origin       h(u,v)=ln(nu2-u2v2+nv2/u2+v2

A:

Q: Use the Fundamental Theorem of Line Integrals to compute the work done F on an object that is moved…

A:

Q: Show that is integrable on [0, 1]. f(x) = {1. 0, +√√xsin(1/x6), x‡0 X 0

A: Given that f(x) = 1 + x sin 1x6,x≠00,x=0 To prove that, f(x) is integrable on [0, 1]

Q: Let f(x): = [2x¹, -ㅠ < x < 0 (5sin(x), 0 ≤ x < ㅠ 음ㅠㅠ 15 - 10 83 – 10 ㅇ ㅠㅠ 25 +10 Evaluate the…

A:

Q: Evaluate the double integral using single applications of Simpson's 1/3 rule 2 (x2. 3y2 + xy³) dx dy…

A:

Q: (a) Let f: R" → R be a function given by f(x₁, 2,...,xn) = x²₁₁x²2x², where n = 1. Show that the…

A:

Q: Let ƒ : [0, 1] → R be the function given by { Is f Riemann integrable? Justify your answer. f(x) = X…

A:

Q: (c) Let f : [0, 2] → IR be defined by 5 if 0 <a < 1 f(x) := {3 if a = 1 2 if 1 < r < 2 Use…

A: Given that: fx=530≤x&lt;1x=121&lt;x≤2 The integral of the given function on [0,2]. I=∫01fxdx+∫12fxdx

Q: et f(x) = 6x? on [(1, 3.5). a) Find L(f, P) and U(f, P) when P 3.5 p) Use calculus to evaluate 6x…

A: In this question, the concept of Riemann Integral is applied. Riemann Integral   To estimate…

Q: Find the singular integral of z = px + qy + p/q

A: Complete integral

Q: Evaluate function given assuming ƒ is integrable on [a, b].

A: We have to calculate,∫abf(x)dx+∫baf(x)dx = ?            ...................(1)As function f(x) is…

Q: Evaluate the integral dy (tan ¹(y/8)) (64+ y²) dy (tan¹(y/8) (64+ y²) -

A:

Q: Is the function f(x) = ( 1, x not equal 1)             ( 0, x equal 1) integrable over the interval…

A: Here, we will show that the given function is integrable. Let us take a point c ∈ (0,1).

Q: 7. If f is continuous on [0, 1] and if [s f(x)x" dx = 0 (n = 0, 1, 2, ...), prove that f(x) = 0 on…

A: We use Weirstrass Theorem

Q: Show that, if f and g are Reimann integrable on [a, b] then f - g is Reimann integrable on [a, b],…

A: We have to prove that following identity.

Q: Refer to V = C[0,1], with the inner product given by an integral. Compute (f.g), where f(t) = 6t-4…

A: inner product

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.