Use the limit definition of definite integral in the form of f(æ) dæ = lim {(#:)Ax, i=1 6 - a and r; = a + iAx, to evaluate the definite integral (4 – 2) dx. n(n+ 1), k = 2 k=1

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A: To prove:

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Q: Express the limit as a definite integral on the giveninterval.

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Q: Express the limit as a definite integral on the given interval. lim > x; In(2 + x;2) Ax, [2, 5] n-…

A: The solution is given below

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A: Given function ,     f(x)= sin x

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Q: Express the limit as a definite integral on the given interval. im. (4(x;") - 3x;JAx, [2, 7] i = 1 7…

A: By limit definition of a definite integral, limn→∞∑i=1nfxi*∆x=∫ab f(x)dx  where ∆x=b-an and…

Q: Express the limit as a definite integral on the given interval. n lim x, In(2 + x;²) Ax, [0, 3] j=1…

A: Given: The limit limn→∞ ∑i=1nxiln2+xi2∆x on interval 0,3 To determine: The interval notation of…

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Q: Express the limit as a definite integral on the given interval. n eši -Ax 4 + X; lim [0, 2] i = 1

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Q: Evaluate the integral as a limit of Riemann Su (3x2-4x+ 1)dx

A: Given:          ∫043x2-4x+1dx here a=0,b=4  Limit of Riemann sum:       ∆x=b-an        ∆x=4-0n=4n…

Q: Use the limit definition of the definite integral to evaluate the following integral. +1

A: Write formula and Put the limit value

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Q: Evaluate the definite integral: x²)dx using limit of the Riemann sum.

A: Consider the provided definite integral, ∫02 2-x2dx Evaluate the provided integration using limit of…

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A: Given:           I=∫-112-3x2dx To Find I using definite integral as a limit of sum:

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Q: So (x²+3x+3) dx Theorem 4% f6) dx= lim f (xi) Ax where Ax= b-a and x= atcAX

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Q: (a) '8 dx 2

A: (a)∫268 dx=[8x]26We know that ∫k dx=kx+C            = 8*6 - 8*2            = 32

Q: 3. Express the given integral as the limit of a Riemann sum but do not evaluate: So (x³ - 6x) dx.

A: Solution:-

Q: xpress the given definite integrals as limit of Riemann sum 5 dx 3 а. b. cOS x dx

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Q: Evaluate the definite integral by the limit definition. L (x² + 4) dx X 31

A: use the limit formula for integration

Q: Express the limit as a definite integral on the given interval. lim Ax [О, 6] 9 + X; i = 1 dx

A: The answer is found in step-2

Q: Express the limit as a definite integral on the given interval. lim. x, In(2 + x?) Ax, [2, 4] n- 00…

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Q: Express the limit as a definite integral on the given interval. lim x; In(1 + x?) Ax, [0, 1] dx

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Q: Q5. Apply the mean value theorem for integrals to evaluate e Lim -dx. 2 + CoS x

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Q: Express the limit as a definite integral on the given interval. lim x, In(2 + x?) Ax, [0, 3] n- 00 i…

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Q: The improper integral f(x)dx can be written as lim f(x)dx + lim_ f" f(x)dx . w--00 w+00 If f(x) =…

A: use the above definition of improper integral

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Q: Use the definition of definite integral (as the limit of a sum) to evaluate | (2x – 1)° dzr.

A: ∫-212x-12 dx Here, a=-2,  b=1  &amp;  fx=2x-12 Now, ∆x=b-an=1--2n=3n

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A: Given that: 1. limn→∞∑i=1nxiln(1+xi2)∆x, [0, 1]2. ∫13(x-4lnx)dx

Q: Express the limit as a definite integral on the given interval. -Дх, [1, 8] lim х? (x;)? + 8 Ax dx

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Q: Express the limit as a definite integral on the given interval. lim Σ Σ x;* -ΔΧ, [1,4] 0-700 (x;*)2…

A: To express the given limit as a definite integral.

Q: 1. Use the definition of the definite integral as the limit of a sum to evaluate (4.x – 2²) dx.

A: The given problem can be solved as-

Q: Express the limit as a definite integral on the given interval. exi Ax 3 + x; lim [0, 4] dx

A: We need convert summation in to integral

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A: Given limn→∞∑i=1ncos(xi)xi∆xi in the interval [2π,5π]We need to express the limit as a  definite…

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A: Let a,b&gt;0 with b&gt;a Then we can write, ∫abf(x) dx as ∫0bf(x) dx - ∫0af(x)dx

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Q: Use the definition of the definite integral as a limit of a sum to evaluate (x² – 4x - 2) dx -2

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Q: Evaluate the definite integral by the limit definition. x dx

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Q: Write the improper integral which is represented by the following limit. 2x dx lim C-2 Show that the…

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Q: Express the limit as a definite integral on the given interval. „m. 2 (3(x;)° – 6x;*]Ax, [2, 6] i =…

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Q: Express the limit as a definite integral on the giv cos Xi Ax, [3x, 4x] lim Xi i = 1 4л dx

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Q: Consider the following theorem. Theorem If f is integrable on [a, b], then FX) dx = lim ° fx;,)ax n-…

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Q: Evaluate the definite integral by the limit definition

A: ∫268dxFormula:∫abcdx=cxab=cb-a   , where c is any constant.